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Glossary of graph theory

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Look upAppendix:Glossary of graph theory in Wiktionary, the free dictionary.

See also:Gallery of named graphs

This is aglossary of graph theory.Graph theory is the study ofgraphs, systems of nodes orvertices connected in pairs by lines oredges.

Symbols

Square brackets [ ]: G[S] is theinduced subgraph of a graphG for vertex subsetS.
Prime symbol ': Theprime symbol is often used to modify notation for graph invariants so that it applies to theline graph instead of the given graph. For instance,α(G) is the independence number of a graph;α′(G) is the matching number of the graph, which equals the independence number of its line graph. Similarly,χ(G) is the chromatic number of a graph;χ ′(G) is the chromatic index of the graph, which equals the chromatic number of its line graph.

A

absorbing: An absorbing set $A {\displaystyle A}$ of a directed graph $G {\displaystyle G}$ is a set of vertices such that for any vertex $v\in G\setminus A$ , there is an edge from $v {\displaystyle v}$ towards a vertex of $A {\displaystyle A}$ .
achromatic: Theachromatic number of a graph is the maximum number of colors in a complete coloring.^[1]
acyclic: 1. A graph is acyclic if it has no cycles. An undirected acyclic graph is the same thing as aforest. An acyclic directed graph, which is a digraph without directed cycles, is often called adirected acyclic graph, especially in computer science.^[2]; 2. Anacyclic coloring of an undirected graph is a proper coloring in which every two color classes induce a forest.^[3]
adjacency matrix: Theadjacency matrix of a graph is a matrix whose rows and columns are both indexed by vertices of the graph, with a one in the cell for rowi and columnj when verticesi andj are adjacent, and a zero otherwise.^[4]
adjacent: 1. The relation between two vertices that are both endpoints of the same edge.^[2]; 2. The relation between two distinct edges that share an end vertex.^[5]
α: For a graphG,α(G) (using the Greek letter alpha) is its independence number (seeindependent), andα′(G) is its matching number (seematching).
alternating: In a graph with a matching, an alternating path is a path whose edges alternate between matched and unmatched edges. An alternating cycle is, similarly, a cycle whose edges alternate between matched and unmatched edges. An augmenting path is an alternating path that starts and ends at unsaturated vertices. A larger matching can be found as thesymmetric difference of the matching and the augmenting path; a matching is maximum if and only if it has no augmenting path.
antichain: In adirected acyclic graph, a subsetS of vertices that are pairwise incomparable, i.e., for any $x\leq y$ inS, there is no directed path fromx toy or fromy tox. Inspired by the notion ofantichains inpartially ordered sets.
anti-edge: Synonym fornon-edge, a pair of non-adjacent vertices.
anti-triangle: A three-vertex independent set, the complement of a triangle.
apex: 1. Anapex graph is a graph in which one vertex can be removed, leaving aplanar subgraph. The removed vertex is called the apex. Ak-apex graph is a graph that can be made planar by the removal ofk vertices.; 2. Synonym foruniversal vertex, a vertex adjacent to all other vertices.
arborescence: Synonym for a rooted and directed tree; seetree.
arc: Seeedge.
arrow: Anordered pair ofvertices, such as anedge in adirected graph. An arrow(x,y) has atailx, aheady, and adirectionfromx toy;y is said to be thedirect successor tox andx thedirect predecessor toy. The arrow(y,x) is theinverted arrow of the arrow(x,y).
articulation point: Avertex in aconnected graph whose removal woulddisconnect the graph. More generally, a vertex whose removal increases the number ofcomponents.
-ary: Ak-ary tree is a rooted tree in which every internal vertex has no more thank children. A 1-ary tree is just a path. A 2-ary tree is also called abinary tree, although that term more properly refers to 2-ary trees in which the children of each node are distinguished as being left or right children (with at most one of each type). Ak-ary tree is said to be complete if every internal vertex has exactlyk children.
augmenting: A special type of alternating path; seealternating.
automorphism: Agraph automorphism is a symmetry of a graph, an isomorphism from the graph to itself.

B

bag: One of the sets of vertices in atree decomposition.
balanced: A bipartite or multipartite graph is balanced if each two subsets of its vertex partition have sizes within one of each other.
ball: A ball (also known as a neighborhood ball or distance ball) is the set of all vertices that are at most distance r from a vertex. More formally, for a given vertex v and radius r, the ball B(v,r) consists of all vertices whose shortest path distance to v is less than or equal to r.
bandwidth: Thebandwidth of a graphG is the minimum, over all orderings of vertices ofG, of the length of the longest edge (the number of steps in the ordering between its two endpoints). It is also one less than the size of the maximum clique in a proper interval completion ofG, chosen to minimize the clique size.
biclique: Synonym forcomplete bipartite graph or complete bipartite subgraph; seecomplete.
biconnected: Usually a synonym for2-vertex-connected, but sometimes includesK₂ though it is not 2-connected. Seeconnected; forbiconnected components, seecomponent.
binding number: The smallest possible ratio of the number of neighbors of a proper subset of vertices to the size of the subset.^[6]
bipartite: Abipartite graph is a graph whose vertices can be divided into two disjoint sets such that the vertices in one set are not connected to each other, but may be connected to vertices in the other set. Put another way, a bipartite graph is a graph with no odd cycles; equivalently, it is a graph that may be properly colored with two colors. Bipartite graphs are often writtenG = (U,V,E) whereU andV are the subsets of vertices of each color. However, unless the graph is connected, it may not have a unique 2-coloring.
biregular: Abiregular graph is abipartite graph in which there are only two different vertex degrees, one for each set of the vertex bipartition.
block: 1. A block of a graphG is a maximal subgraph which is either an isolated vertex, a bridge edge, or a 2-connected subgraph. If a block is 2-connected, every pair of vertices in it belong to a common cycle. Every edge of a graph belongs in exactly one block.; 2. The block graph of a graphG is another graph whose vertices are the blocks ofG, with an edge connecting two vertices when the corresponding blocks share an articulation point; that is, it is the intersection graph of the blocks ofG. The block graph of any graph is aforest.; 3. The block-cut (or block-cutpoint) graph of a graphG is a bipartite graph where one partite set consists of the cut-vertices ofG, and the other has a vertex $b_{i}$ for each block $B_{i}$ ofG. WhenG is connected, its block-cutpoint graph is a tree.; 4. Ablock graph (also called a clique tree if connected, and sometimes erroneously called a Husimi tree) is a graph all of whose blocks are complete graphs. Aforest is a block graph; so in particular the block graph of any graph is a block graph, and every block graph may be constructed as the block graph of a graph.
bond: Aminimal cut-set: a set of edges whose removal disconnects the graph, for which no proper subset has the same property.
book: 1. Abook, book graph, or triangular book is a complete tripartite graphK_1,1,n; a collection ofn triangles joined at a shared edge.; 2. Another type of graph, also called a book, or a quadrilateral book, is a collection of4-cycles joined at a shared edge; the Cartesian product of a star with an edge.; 3. Abook embedding is an embedding of a graph onto a topological book, a space formed by joining a collection of half-planes along a shared line. Usually, the vertices of the embedding are required to be on the line, which is called the spine of the embedding, and the edges of the embedding are required to lie within a single half-plane, one of the pages of the book.
boundary: 1. In agraph embedding, a boundary walk is the subgraph containing all incident edges and vertices to aface.
bramble: Abramble is a collection of mutually touching connected subgraphs, where two subgraphs touch if they share a vertex or each includes one endpoint of an edge. The order of a bramble is the smallest size of a set of vertices that has a nonempty intersection with all of the subgraphs. The treewidth of a graph is the maximum order of any of its brambles.
branch: A path of degree-two vertices, ending at vertices whose degree is unequal to two.^[7]
branch-decomposition: Abranch-decomposition ofG is a hierarchical clustering of the edges ofG, represented by an unrooted binary tree with its leaves labeled by the edges ofG. The width of a branch-decomposition is the maximum, over edgese of this binary tree, of the number of shared vertices between the subgraphs determined by the edges ofG in the two subtrees separated bye. The branchwidth ofG is the minimum width of any branch-decomposition ofG.
branchwidth: Seebranch-decomposition.
bridge: 1. Abridge, isthmus, or cut edge is an edge whose removal would disconnect the graph. A bridgeless graph is one that has no bridges; equivalently, a 2-edge-connected graph.; 2. A bridge of a subgraphH is a maximal connected subgraph separated from the rest of the graph byH. That is, it is a maximal subgraph that is edge-disjoint fromH and in which each two vertices and edges belong to a path that is internally disjoint fromH.H may be a set of vertices. A chord is a one-edge bridge. Inplanarity testing,H is a cycle and aperipheral cycle is a cycle with at most one bridge; it must be a face boundary in any planar embedding of its graph.; 3. A bridge of a cycle can also mean a path that connects two vertices of a cycle but is shorter than either of the paths in the cycle connecting the same two vertices. Abridged graph is a graph in which every cycle of four or more vertices has a bridge.
bridgeless: Abridgeless or isthmus-free graph is a graph that has no bridge edges (i.e., isthmi); that is, each connected component is a2-edge-connected graph.
butterfly: 1. Thebutterfly graph has five vertices and six edges; it is formed by two triangles that share a vertex.; 2. The butterfly network is a graph used as a network architecture in distributed computing, closely related to thecube-connected cycles.

C

C: C_n is ann-vertexcycle graph; seecycle.
cactus: Acactus graph, cactus tree, cactus, or Husimi tree is a connected graph in which each edge belongs to at most one cycle. Its blocks are cycles or single edges. If, in addition, each vertex belongs to at most two blocks, then it is called a Christmas cactus.
cage: Acage is a regular graph with the smallest possible order for its girth.
canonical
canonization: Acanonical form of a graph is an invariant such that two graphs have equal invariants if and only if they are isomorphic. Canonical forms may also be called canonical invariants or complete invariants, and are sometimes defined only for the graphs within a particular family of graphs.Graph canonization is the process of computing a canonical form.
card: A graph formed from a given graph by deleting one vertex, especially in the context of thereconstruction conjecture. See alsodeck, the multiset of all cards of a graph.
carving width: Carving width is a notion of graph width analogous to branchwidth, but using hierarchical clusterings of vertices instead of hierarchical clusterings of edges.
caterpillar: Acaterpillar tree or caterpillar is a tree in which the internal nodes induce a path.
center: Thecenter of a graph is the set of vertices of minimumeccentricity.
centroid: Acentroid of a tree is a vertexv such that if rooted atv, no other vertex has subtree size greater than half the size of the tree.
chain: 1. Synonym forwalk.; 2. When applying methods fromalgebraic topology to graphs, an element of achain complex, namely a set of vertices or a set of edges.
Cheeger constant: Seeexpansion.
cherry: A cherry is a path on three vertices.^[8]
χ: χ(G) (using the Greek letter chi) is the chromatic number ofG andχ ′(G) is its chromatic index; seechromatic andcoloring.
child: In a rooted tree, a child of a vertexv is a neighbor ofv along an outgoing edge, one that is directed away from the root.
chord
chordal: 1. A chord of a cycle is an edge that does not belong to the cycle, for which both endpoints belong to the cycle.; 2. Achordal graph is a graph in which every cycle of four or more vertices has a chord, so the only induced cycles are triangles.; 3. Astrongly chordal graph is a chordal graph in which every cycle of length six or more has an odd chord.; 4. Achordal bipartite graph is not chordal (unless it is a forest); it is a bipartite graph in which every cycle of six or more vertices has a chord, so the only induced cycles are 4-cycles.; 5. Achord of a circle is a line segment connecting two points on the circle; theintersection graph of a collection of chords is called acircle graph.
chromatic: Having to do with coloring; seecolor. Chromatic graph theory is the theory of graph coloring. Thechromatic numberχ(G) is the minimum number of colors needed in a proper coloring ofG.χ ′(G) is thechromatic index ofG, the minimum number of colors needed in a properedge coloring ofG.
choosable
choosability: A graph isk-choosable if it has alist coloring whenever each vertex has a list ofk available colors. The choosability of the graph is the smallestk for which it isk-choosable.
circle: Acircle graph is theintersection graph of chords of a circle.
circuit: A circuit may refer to a closed trail or an element of thecycle space (an Eulerian spanning subgraph). Thecircuit rank of a graph is the dimension of its cycle space.
circumference: Thecircumference of a graph is the length of its longest simple cycle. The graph is Hamiltonian if and only if its circumference equals its order.
class: 1. Aclass of graphs or family of graphs is a (usually infinite) collection of graphs, often defined as the graphs having some specific property. The word "class" is used rather than "set" because, unless special restrictions are made (such as restricting the vertices to be drawn from a particular set, and defining edges to be sets of two vertices) classes of graphs are usually not sets when formalized using set theory.; 2. A color class of a colored graph is the set of vertices or edges having one particular color.; 3. In the context ofVizing's theorem, on edge coloring simple graphs, a graph is said to be of class one if its chromatic index equals its maximum degree, and class two if its chromatic index equals one plus the degree. According to Vizing's theorem, all simple graphs are either of class one or class two.
claw: Aclaw is a tree with one internal vertex and three leaves, or equivalently the complete bipartite graphK_1,3. Aclaw-free graph is a graph that does not have an induced subgraph that is a claw.
clique: Aclique is a set of mutually adjacent vertices (or the complete subgraph induced by that set). Sometimes a clique is defined as a maximal set of mutually adjacent vertices (or maximal complete subgraph), one that is not part of any larger such set (or subgraph). Ak-clique is a clique of orderk. Theclique numberω(G) of a graphG is the order of its largest clique. Theclique graph of a graphG is theintersection graph of the maximal cliques inG. See alsobiclique, a complete bipartite subgraph.
clique tree: A synonym for ablock graph.
clique-width: Theclique-width of a graphG is the minimum number of distinct labels needed to constructG by operations that create a labeled vertex, form the disjoint union of two labeled graphs, add an edge connecting all pairs of vertices with given labels, or relabel all vertices with a given label. The graphs of clique-width at most2 are exactly thecographs.
closed: 1. A closed neighborhood is one that includes its central vertex; seeneighbourhood.; 2. A closed walk is one that starts and ends at the same vertex; seewalk.; 3. A graph is transitively closed if it equals its own transitive closure; seetransitive.; 4. A graph property is closed under some operation on graphs if, whenever the argument or arguments to the operation have the property, then so does the result. For instance, hereditary properties are closed under induced subgraphs; monotone properties are closed under subgraphs; and minor-closed properties are closed under minors.
closure: 1. For the transitive closure of a directed graph, seetransitive.; 2. A closure of a directed graph is a set of vertices that have no outgoing edges to vertices outside the closure. For instance, a sink is a one-vertex closure. Theclosure problem is the problem of finding a closure of minimum or maximum weight.
co-: This prefix has various meanings usually involvingcomplement graphs. For instance, acograph is a graph produced by operations that include complementation; acocoloring is a coloring in which each vertex induces either an independent set (as in proper coloring) or a clique (as in a coloring of the complement).
color
coloring: 1. Agraph coloring is a labeling of the vertices of a graph by elements from a given set of colors, or equivalently a partition of the vertices into subsets, called "color classes", each of which is associated with one of the colors.; 2. Some authors use "coloring", without qualification, to mean a proper coloring, one that assigns different colors to the endpoints of each edge. In graph coloring, the goal is to find a proper coloring that uses as few colors as possible; for instance,bipartite graphs are the graphs that have colorings with only two colors, and thefour color theorem states that everyplanar graph can be colored with at most four colors. A graph is said to bek-colored if it has been (properly) colored withk colors, andk-colorable ork-chromatic if this is possible.; 3. Many variations of coloring have been studied, includingedge coloring (coloring edges so that no two edges with the same endpoint share a color),list coloring (proper coloring with each vertex restricted to a subset of the available colors),acyclic coloring (every 2-colored subgraph is acyclic), co-coloring (every color class induces an independent set or a clique),complete coloring (every two color classes share an edge), andtotal coloring (both edges and vertices are colored).; 4. The coloring number of a graph is one plus thedegeneracy. It is so called because applying a greedy coloring algorithm to a degeneracy ordering of the graph uses at most this many colors.
comparability: An undirected graph is acomparability graph if its vertices are the elements of apartially ordered set and two vertices are adjacent when they are comparable in the partial order. Equivalently, a comparability graph is a graph that has a transitive orientation. Many other classes of graphs can be defined as the comparability graphs of special types of partial order.
complement: Thecomplement graph ${\bar {G}}$ of a simple graphG is another graph on the same vertex set asG, with an edge for each two vertices that are not adjacent inG.
complete: 1. Acomplete graph is one in which every two vertices are adjacent: all edges that could exist are present. A complete graph withn vertices is often denotedK_n. Acomplete bipartite graph is one in which every two vertices on opposite sides of the partition of vertices are adjacent. A complete bipartite graph witha vertices on one side of the partition andb vertices on the other side is often denotedK_a,b. The same terminology and notation has also been extended tocomplete multipartite graphs, graphs in which the vertices are divided into more than two subsets and every pair of vertices in different subsets are adjacent; if the numbers of vertices in the subsets area,b,c, ... then this graph is denotedK_{a,b,c, ...}.; 2. A completion of a given graph is a supergraph that has some desired property. For instance, achordal completion is a supergraph that is a chordal graph.; 3. A complete matching is a synonym for aperfect matching; seematching.; 4. Acomplete coloring is a proper coloring in which each pairs of colors is used for the endpoints of at least one edge. Every coloring with a minimum number of colors is complete, but there may exist complete colorings with larger numbers of colors. Theachromatic number of a graph is the maximum number of colors in a complete coloring.; 5. A complete invariant of a graph is a synonym for a canonical form, an invariant that has different values for non-isomorphic graphs.
component: Aconnected component of a graph is a maximal connected subgraph. The term is also used for maximal subgraphs or subsets of a graph's vertices that have some higher order of connectivity, includingbiconnected components,triconnected components, andstrongly connected components.
condensation: Thecondensation of a directed graphG is a directed acyclic graph with one vertex for each strongly connected component ofG, and an edge connecting pairs of components that contain the two endpoints of at least one edge inG.
cone: A graph that contains auniversal vertex.
connect: Cause to beconnected.
connected: Aconnected graph is one in which each pair of vertices forms the endpoints of a path. Higher forms of connectivity include strong connectivity in directed graphs (for each two vertices there are paths from one to the other in both directions),k-vertex-connected graphs (removing fewer thank vertices cannot disconnect the graph), andk-edge-connected graphs (removing fewer thank edges cannot disconnect the graph).
connected component: Synonym forcomponent.
contraction: Edge contraction is an elementary operation that removes an edge from a graph while merging the two vertices that it previously joined. Vertex contraction (sometimes called vertex identification) is similar, but the two vertices are not necessarily connected by an edge. Path contraction occurs upon the set of edges in a path that contract to form a single edge between the endpoints of the path. The inverse of edge contraction is vertex splitting.
converse: The converse graph is a synonym for the transpose graph; seetranspose.
core: 1. Ak-core is the induced subgraph formed by removing all vertices of degree less thank, and all vertices whose degree becomes less thank after earlier removals. Seedegeneracy.; 2. Acore is a graphG such that everygraph homomorphism fromG to itself is an isomorphism.; 3. Thecore of a graphG is a minimal graphH such that there exist homomorphisms fromG toH and vice versa.H is unique up to isomorphism. It can be represented as an induced subgraph ofG, and is a core in the sense that all of its self-homomorphisms are isomorphisms.; 4. In the theory of graph matchings, the core of a graph is an aspect of itsDulmage–Mendelsohn decomposition, formed as the union of all maximum matchings.
cotree: 1. The complement of aspanning tree.; 2. A rooted tree structure used to describe acograph, in which each cograph vertex is a leaf of the tree, each internal node of the tree is labeled with 0 or 1, and two cograph vertices are adjacent if and only if their lowest common ancestor in the tree is labeled 1.
cover: Avertex cover is a set of vertices incident to every edge in a graph. Anedge cover is a set of edges incident to every vertex in a graph. A set of subgraphs of a graph covers that graph if itsunion – taken vertex-wise and edge-wise – is equal to the graph.
critical: A critical graph for a given property is a graph that has the property but such that every subgraph formed by deleting a single vertex does not have the property. For instance, afactor-critical graph is one that has a perfect matching (a 1-factor) for every vertex deletion, but (because it has an odd number of vertices) has no perfect matching itself. Comparehypo-, used for graphs which do not have a property but for which every one-vertex deletion does.
cube
cubic: 1. Cube graph, the eight-vertex graph of the vertices and edges of a cube.; 2. Hypercube graph, a higher-dimensional generalization of the cube graph.; 3. Folded cube graph, formed from a hypercube by adding a matching connecting opposite vertices.; 4. Halved cube graph, thehalf-square of a hypercube graph.; 5. Partial cube, a distance-preserving subgraph of a hypercube.; 6. The cube of a graphG is thegraph powerG³.; 7. Cubic graph, another name for a3-regular graph, one in which each vertex has three incident edges.; 8. Cube-connected cycles, a cubic graph formed by replacing each vertex of a hypercube by a cycle.
cut
cut-set: Acut is a partition of the vertices of a graph into two subsets, or the set (also known as a cut-set) of edges that span such a partition, if that set is non-empty. An edge is said to span the partition if it has endpoints in both subsets. Thus, the removal of a cut-set from a connected graph disconnects it.
cut point: Seearticulation point.
cut space: Thecut space of a graph is aGF(2)-vector space having thecut-sets of the graph as its elements andsymmetric difference of sets as its vector addition operation.
cycle: 1. Acycle may be either a kind of graph or a kind ofwalk. As a walk it may be either be a closed walk (also called atour) or more usually a closed walk without repeated vertices and consequently edges (also called a simple cycle). In the latter case it is usually regarded as a graph, i.e., the choices of first vertex and direction are usually considered unimportant; that is,cyclic permutations and reversals of the walk produce the same cycle. Important special types of cycle includeHamiltonian cycles,induced cycles,peripheral cycles, and the shortest cycle, which defines thegirth of a graph. Ak-cycle is a cycle of lengthk; for instance a2-cycle is adigon and a3-cycle is a triangle. Acycle graph is a graph that is itself a simple cycle; a cycle graph withn vertices is commonly denotedC_n.; 2. Thecycle space is avector space generated by the simple cycles in a graph, often over the field of 2 elements but also over other fields.

D

DAG: Abbreviation fordirected acyclic graph, a directed graph without any directed cycles.
deck: The multiset of graphs formed from a single graphG by deleting a single vertex in all possible ways, especially in the context of thereconstruction conjecture. An edge-deck is formed in the same way by deleting a single edge in all possible ways. The graphs in a deck are also calledcards. See alsocritical (graphs that have a property that is not held by any card) andhypo- (graphs that do not have a property that is held by all cards).
decomposition: Seetree decomposition,path decomposition, orbranch-decomposition.
degenerate
degeneracy: Ak-degenerate graph is an undirected graph in which every induced subgraph has minimum degree at mostk. Thedegeneracy of a graph is the smallestk for which it isk-degenerate. A degeneracy ordering is an ordering of the vertices such that each vertex has minimum degree in the induced subgraph of it and all later vertices; in a degeneracy ordering of ak-degenerate graph, every vertex has at mostk later neighbours. Degeneracy is also known as thek-core number, width, and linkage, and one plus the degeneracy is also called the coloring number or Szekeres–Wilf number.k-degenerate graphs have also been calledk-inductive graphs.
degree: 1. Thedegree of a vertex in a graph is its number of incident edges.^[2] The degree of a graphG (or its maximum degree) is the maximum of the degrees of its vertices, often denotedΔ(G); the minimum degree ofG is the minimum of its vertex degrees, often denotedδ(G). Degree is sometimes calledvalency; the degree ofv inG may be denotedd_G(v),d(G), ordeg(v). The total degree is the sum of the degrees of all vertices; by thehandshaking lemma it is an even number. Thedegree sequence is the collection of degrees of all vertices, in sorted order from largest to smallest. In a directed graph, one may distinguish the in-degree (number of incoming edges) and out-degree (number of outgoing edges).^[2]; 2. The homomorphism degree of a graph is a synonym for itsHadwiger number, the order of the largest clique minor.
Δ,δ: Δ(G) (using the Greek letter delta) is the maximum degree of a vertex inG, andδ(G) is the minimum degree; seedegree.
density: In a graph ofn nodes, the density is the ratio of the number of edges of the graph to the number of edges in a complete graph onn nodes. Seedense graph.
depth: The depth of a node in a rooted tree is the number of edges in the path from the root to the node. For instance, the depth of the root is 0 and the depth of any one of its adjacent nodes is 1. It is the level of a node minus one. Note, however, that some authors instead usedepth as a synonym for thelevel of a node.^[9]
diameter: Thediameter of a connected graph is the maximum length of ashortest path. That is, it is the maximum of the distances between pairs of vertices in the graph. If the graph has weights on its edges, then its weighted diameter measures path length by the sum of the edge weights along a path, while the unweighted diameter measures path length by the number of edges.For disconnected graphs, definitions vary: the diameter may be defined as infinite, or as the largest diameter of a connected component, or it may be undefined.
diamond: Thediamond graph is an undirected graph with four vertices and five edges.
diconnected: Stronglyconnected. (Not to be confused withdisconnected)
digon: Adigon is a simple cycle of length two in a directed graph or a multigraph. Digons cannot occur insimple undirected graphs as they require repeating the same edge twice, which violates the definition ofsimple.
digraph: Synonym fordirected graph.^[2]
dipath: Seedirected path.
direct predecessor: The tail of a directed edge whose head is the given vertex.
direct successor: The head of a directed edge whose tail is the given vertex.
directed: Adirected graph is one in which the edges have a distinguished direction, from one vertex to another.^[2] In amixed graph, a directed edge is again one that has a distinguished direction; directed edges may also be called arcs or arrows.
directed arc: Seearrow.
directed edge: Seearrow.
directed line: Seearrow.
directed path: Apath in which all theedges have the samedirection. If a directed path leads fromvertexx to vertexy,x is apredecessor ofy,y is asuccessor ofx, andy is said to bereachable fromx.
direction: 1. Theasymmetric relation between twoadjacent vertices in agraph, represented as anarrow.; 2. The asymmetric relation between two vertices in adirected path.
disconnect: Cause to bedisconnected.
disconnected: Notconnected.
disjoint: 1. Two subgraphs are edge disjoint if they share no edges, and vertex disjoint if they share no vertices.; 2. The disjoint union of two or more graphs is a graph whose vertex and edge sets are thedisjoint unions of the corresponding sets.
dissociation number: A subset of vertices in a graphG is calleddissociation if it induces asubgraph with maximumdegree 1.
distance: Thedistance between any two vertices in a graph is the length of the shortest path having the two vertices as its endpoints.
domatic: A domatic partition of a graph is a partition of the vertices into dominating sets. Thedomatic number of the graph is the maximum number of dominating sets in such a partition.
dominating: Adominating set is a set of vertices that includes or is adjacent to every vertex in the graph; not to be confused with a vertex cover, a vertex set that is incident to all edges in the graph. Important special types of dominating sets include independent dominating sets (dominating sets that are also independent sets) and connected dominating sets (dominating sets that induced connected subgraphs). A single-vertex dominating set may also be called a universal vertex. The domination number of a graph is the number of vertices in the smallest dominating set.
dual: Adual graph of a plane graphG is a graph that has a vertex for each face ofG.

E

E: E(G) is the edge set ofG; seeedge set.
ear: An ear of a graph is a path whose endpoints may coincide but in which otherwise there are no repetitions of vertices or edges.
ear decomposition: Anear decomposition is a partition of the edges of a graph into a sequence of ears, each of whose endpoints (after the first one) belong to a previous ear and each of whose interior points do not belong to any previous ear. An open ear is a simple path (an ear without repeated vertices), and an open ear decomposition is an ear decomposition in which each ear after the first is open; a graph has an open ear decomposition if and only if it is biconnected. An ear is odd if it has an odd number of edges, and an odd ear decomposition is an ear decomposition in which each ear is odd; a graph has an odd ear decomposition if and only if it is factor-critical.
eccentricity: The eccentricity of a vertex is the farthest distance from it to any other vertex.
edge: An edge is (together with vertices) one of the two basic units out of which graphs are constructed. Each edge has two (or in hypergraphs, more) vertices to which it is attached, called its endpoints. Edges may be directed or undirected; undirected edges are also called lines and directed edges are also called arcs or arrows. In an undirectedsimple graph, an edge may be represented as the set of its vertices, and in a directed simple graph it may be represented as an ordered pair of its vertices. An edge that connects verticesx andy is sometimes writtenxy.
edge cut: A set ofedges whose removaldisconnects thegraph. A one-edge cut is called abridge,isthmus, orcut edge.
edge set: The set of edges of a given graphG, sometimes denoted byE(G).
edgeless graph: Theedgeless graph or totally disconnected graph on a given set of vertices is the graph that has no edges. It is sometimes called the empty graph, but this term can also refer to a graph with no vertices.
embedding: Agraph embedding is a topological representation of a graph as a subset of a topological space with each vertex represented as a point, each edge represented as a curve having the endpoints of the edge as endpoints of the curve, and no other intersections between vertices or edges. Aplanar graph is a graph that has such an embedding onto the Euclidean plane, and atoroidal graph is a graph that has such an embedding onto a torus. Thegenus of a graph is the minimum possible genus of a two-dimensionalmanifold onto which it can be embedded.
empty graph: 1. Anedgeless graph on a nonempty set of vertices.; 2. Theorder-zero graph, a graph with no vertices and no edges.
end: Anend of an infinite graph is an equivalence class of rays, where two rays are equivalent if there is a third ray that includes infinitely many vertices from both of them.
endpoint: One of the two vertices joined by a given edge, or one of the first or last vertex of a walk, trail or path. The first endpoint of a given directed edge is called thetail and the second endpoint is called thehead.
enumeration: Graph enumeration is the problem of counting the graphs in a given class of graphs, as a function of their order. More generally, enumeration problems can refer either to problems of counting a certain class of combinatorial objects (such as cliques, independent sets, colorings, or spanning trees), or of algorithmically listing all such objects.
Eulerian: AnEulerian path is a walk that uses every edge of a graph exactly once. An Eulerian circuit (also called an Eulerian cycle or an Euler tour) is a closed walk that uses every edge exactly once. An Eulerian graph is a graph that has an Eulerian circuit. For an undirected graph, this means that the graph is connected and every vertex has even degree. For a directed graph, this means that the graph is strongly connected and every vertex has in-degree equal to the out-degree. In some cases, the connectivity requirement is loosened, and a graph meeting only the degree requirements is called Eulerian.
even: Divisible by two; for instance, an even cycle is a cycle whose length is even.
expander: Anexpander graph is a graph whose edge expansion, vertex expansion, or spectral expansion is bounded away from zero.
expansion: 1. The edge expansion, isoperimetric number, orCheeger constant of a graphG is the minimum ratio, over subsetsS of at most half of the vertices ofG, of the number of edges leavingS to the number of vertices inS.; 2. The vertex expansion, vertex isoperimetric number, or magnification of a graphG is the minimum ratio, over subsetsS of at most half of the vertices ofG, of the number of vertices outside but adjacent toS to the number of vertices inS.; 3. The unique neighbor expansion of a graphG is the minimum ratio, over subsets of at most half of the vertices ofG, of the number of vertices outsideS but adjacent to a unique vertex inS to the number of vertices inS.; 4. The spectral expansion of ad-regular graphG is thespectral gap between the largest eigenvalued of its adjacency matrix and the second-largest eigenvalue.; 5. A family of graphs hasbounded expansion if all itsr-shallow minors have a ratio of edges to vertices bounded by a function ofr, and polynomial expansion if the function ofr is a polynomial.

F

face: In aplane graph orgraph embedding, a connected component of the subset of the plane or surface of the embedding that is disjoint from the graph. For an embedding in the plane, all but one face will be bounded; the one exceptional face that extends to infinity is called the outer (or infinite) face.
factor: A factor of a graph is a spanning subgraph: a subgraph that includes all of the vertices of the graph. The term is primarily used in the context of regular subgraphs: ak-factor is a factor that isk-regular. In particular, a1-factor is the same thing as a perfect matching. Afactor-critical graph is a graph for which deleting any one vertex produces a graph with a1-factor.
factorization: Agraph factorization is a partition of the edges of the graph into factors; ak-factorization is a partition intok-factors. For instance a1-factorization is an edge coloring with the additional property that each vertex is incident to an edge of each color.
family: A synonym forclass.
finite: A graph is finite if it has a finite number of vertices and a finite number of edges. Many sources assume that all graphs are finite without explicitly saying so. A graph is locally finite if each vertex has a finite number of incident edges. An infinite graph is a graph that is not finite: it has infinitely many vertices, infinitely many edges, or both.
first order: The first orderlogic of graphs is a form of logic in which variables represent vertices of a graph, and there exists a binary predicate to test whether two vertices are adjacent. To be distinguished from second order logic, in which variables can also represent sets of vertices or edges.
-flap: For a set of verticesX, anX-flap is a connected component of the induced subgraph formed by deletingX. The flap terminology is commonly used in the context ofhavens, functions that map small sets of vertices to their flaps. See also thebridge of a cycle, which is either a flap of the cycle vertices or a chord of the cycle.
forbidden: Aforbidden graph characterization is a characterization of a family of graphs as being the graphs that do not have certain other graphs as subgraphs, induced subgraphs, or minors. IfH is one of the graphs that does not occur as a subgraph, induced subgraph, or minor, thenH is said to be forbidden.
forcing graph: Aforcing graph is a graphH such that evaluating the subgraph density ofH in the graphs of a graph sequenceG(n) is sufficient to test whether that sequence isquasi-random.
forest: Aforest is an undirected graph without cycles (a disjoint union of unrooted trees), or a directed graph formed as a disjoint union of rooted trees.
free edge: Anedge which is not in amatching.
free vertex: 1. Avertex not on a matchededge in amatching; 2. A vertex which has not been matched.
Frucht: 1. Robert Frucht; 2. TheFrucht graph, one of the two smallest cubic graphs with no nontrivial symmetries.; 3. Frucht's theorem that every finite group is the group of symmetries of a finite graph.
full: Synonym forinduced.
functional graph: Afunctional graph is a directed graph where every vertex has out-degree one. Equivalently, a functional graph is a maximal directed pseudoforest.

G

G: A variable often used to denote a graph.
genus: The genus of a graph is the minimum genus of a surface onto which it can be embedded; seeembedding.
geodesic: As a noun, a geodesic is a synonym for ashortest path. When used as an adjective, it means related to shortest paths or shortest path distances.
giant: In the theory ofrandom graphs, a giant component is a connected component that contains a constant fraction of the vertices of the graph. In standard models of random graphs, there is typically at most one giant component.
girth: Thegirth of a graph is the length of its shortest cycle.
graph: The fundamental object of study in graph theory, a system of vertices connected in pairs by edges. Often subdivided intodirected graphs orundirected graphs according to whether the edges have an orientation or not.Mixed graphs include both types of edges.
greedy: Produced by agreedy algorithm. For instance, agreedy coloring of a graph is a coloring produced by considering the vertices in some sequence and assigning each vertex the first available color.
Grötzsch: 1. Herbert Grötzsch; 2. TheGrötzsch graph, the smallest triangle-free graph requiring four colors in any proper coloring.; 3. Grötzsch's theorem that triangle-free planar graphs can always be colored with at most three colors.
Grundy number: 1. TheGrundy number of a graph is the maximum number of colors produced by agreedy coloring, with a badly-chosen vertex ordering.

H

H: A variable often used to denote a graph, especially when another graph has already been denoted byG.
H-coloring: AnH-coloring of a graphG (whereH is also a graph) is a homomorphism fromH toG.
H-free: A graph isH-free if it does not have an induced subgraph isomorphic toH, that is, ifH is a forbidden induced subgraph. TheH-free graphs are the family of all graphs (or, often, all finite graphs) that areH-free.^[10] For instance thetriangle-free graphs are the graphs that do not have atriangle graph as a subgraph. The property of beingH-free is always hereditary. A graph isH-minor-free if it does not have a minor isomorphic toH.
Hadwiger: 1. Hugo Hadwiger; 2. TheHadwiger number of a graph is the order of the largest complete minor of the graph. It is also called the contraction clique number or the homomorphism degree.; 3. TheHadwiger conjecture is the conjecture that the Hadwiger number is never less than the chromatic number.
Hamiltonian: AHamiltonian path or Hamiltonian cycle is a simple spanning path or simple spanning cycle: it covers all of the vertices in the graph exactly once. A graph is Hamiltonian if it contains a Hamiltonian cycle, and traceable if it contains a Hamiltonian path.
haven: Ak-haven is a function that maps every setX of fewer thank vertices to one of its flaps, often satisfying additional consistency conditions. The order of a haven is the numberk. Havens can be used to characterize the treewidth of finite graphs and the ends and Hadwiger numbers of infinite graphs.
height: 1. Theheight of a node in a rooted tree is the number of edges in a longest path, going away from the root (i.e. its nodes have strictly increasing depth), that starts at that node and ends at a leaf.; 2. Theheight of a rooted tree is the height of its root. That is, theheight of a tree is the number of edges in a longest possible path, going away from the root, that starts at the root and ends at a leaf.; 3. Theheight of adirected acyclic graph is the maximum length of a directed path in this graph.
hereditary: Ahereditary property of graphs is a property that is closed under induced subgraphs: ifG has a hereditary property, then so must every induced subgraph ofG. Comparemonotone (closed under all subgraphs) orminor-closed (closed under minors).
hexagon: A simple cycle consisting of exactly six edges and six vertices.
hole: A hole is an induced cycle of length four or more. An odd hole is a hole of odd length. An anti-hole is an induced subgraph of order four whose complement is a cycle; equivalently, it is a hole in the complement graph. This terminology is mainly used in the context of perfect graphs, which are characterized by thestrong perfect graph theorem as being the graphs with no odd holes or odd anti-holes. The hole-free graphs are the same as thechordal graphs.
homomorphic equivalence: Two graphs arehomomorphically equivalent if there exist two homomorphisms, one from each graph to the other graph.
homomorphism: 1. Agraph homomorphism is a mapping from the vertex set of one graph to the vertex set of another graph that maps adjacent vertices to adjacent vertices. This type of mapping between graphs is the one that is most commonly used in category-theoretic approaches to graph theory. A proper graph coloring can equivalently be described as a homomorphism to a complete graph.; 2. The homomorphism degree of a graph is a synonym for itsHadwiger number, the order of the largest clique minor.
hyperarc: A directedhyperedge having a source and target set.
hyperedge: Anedge in ahypergraph, having any number of endpoints, in contrast to the requirement that edges of graphs have exactly two endpoints.
hypercube: Ahypercube graph is a graph formed from the vertices and edges of a geometrichypercube.
hypergraph: Ahypergraph is a generalization of a graph in which each edge (called a hyperedge in this context) may have more than two endpoints.
hypo-: This prefix, in combination with a graph property, indicates a graph that does not have the property but such that every subgraph formed by deleting a single vertex does have the property. For instance, ahypohamiltonian graph is one that does not have a Hamiltonian cycle, but for which every one-vertex deletion produces a Hamiltonian subgraph. Comparecritical, used for graphs which have a property but for which every one-vertex deletion does not.^[11]

I

in-degree: The number of incoming edges in a directed graph; seedegree.
incidence: Anincidence in a graph is a vertex-edge pair such that the vertex is an endpoint of the edge.
incidence matrix: Theincidence matrix of a graph is a matrix whose rows are indexed by vertices of the graph, and whose columns are indexed by edges, with a one in the cell for rowi and columnj when vertexi and edgej are incident, and a zero otherwise.
incident: The relation between an edge and one of its endpoints.^[2]
incomparability: An incomparability graph is the complement of acomparability graph; seecomparability.
independent: 1. Anindependent set is a set of vertices that induces an edgeless subgraph. It may also be called a stable set or a coclique. Theindependence numberα(G) is the size of themaximum independent set.; 2. In thegraphic matroid of a graph, a subset of edges is independent if the corresponding subgraph is a tree or forest. In thebicircular matroid, a subset of edges is independent if the corresponding subgraph is apseudoforest.
indifference: Anindifference graph is another name for a proper interval graph or unit interval graph; seeproper.
induced: Aninduced subgraph or full subgraph of a graph is a subgraph formed from a subset of vertices and from all of the edges that have both endpoints in the subset. Special cases includeinduced paths andinduced cycles, induced subgraphs that are paths or cycles.
inductive: Synonym fordegenerate.
infinite: An infinite graph is one that is not finite; seefinite.
internal: A vertex of a path or tree is internal if it is not a leaf; that is, if its degree is greater than one. Two paths are internally disjoint (some people call itindependent) if they do not have any vertex in common, except the first and last ones.
intersection: 1. The intersection of two graphs is their largest common subgraph, the graph formed by the vertices and edges that belong to both graphs.; 2. Anintersection graph is a graph whose vertices correspond to sets or geometric objects, with an edge between two vertices exactly when the corresponding two sets or objects have a nonempty intersection. Several classes of graphs may be defined as the intersection graphs of certain types of objects, for instancechordal graphs (intersection graphs of subtrees of a tree),circle graphs (intersection graphs of chords of a circle),interval graphs (intersection graphs of intervals of a line),line graphs (intersection graphs of the edges of a graph), andclique graphs (intersection graphs of the maximal cliques of a graph). Every graph is an intersection graph for some family of sets, and this family is called an intersection representation of the graph. Theintersection number of a graphG is the minimum total number of elements in any intersection representation ofG.
interval: 1. Aninterval graph is anintersection graph ofintervals of a line.; 2. The interval[u,v] in a graph is the union of all shortest paths fromu tov.; 3. Interval thickness is a synonym forpathwidth.
invariant: A synonym ofproperty.
inverted arrow: Anarrow with an oppositedirection compared to another arrow. The arrow(y,x) is the inverted arrow of the arrow(x,y).
isolated: An isolated vertex of a graph is a vertex whose degree is zero, that is, a vertex with no incident edges.^[2]
isomorphic: Two graphs are isomorphic if there is an isomorphism between them; seeisomorphism.
isomorphism: Agraph isomorphism is a one-to-one incidence preserving correspondence of the vertices and edges of one graph to the vertices and edges of another graph. Two graphs related in this way are said to be isomorphic.
isoperimetric: Seeexpansion.
isthmus: Synonym forbridge, in the sense of an edge whose removal disconnects the graph.

J

join: Thejoin of two graphs is formed from theirdisjoint union by adding an edge from each vertex of one graph to each vertex of the other. Equivalently, it is the complement of the disjoint union of the complements.

K

K: For the notation for complete graphs, complete bipartite graphs, and complete multipartite graphs, seecomplete.
κ: κ(G) (using the Greek letter kappa) can refer to thevertex connectivity ofG or to theclique number ofG.
kernel: A kernel of a directed graph is a set of vertices which is bothstable andabsorbing.
knot: An inescapable section of adirected graph. Seeknot (mathematics) andknot theory.

L

L: L(G) is theline graph ofG; seeline.
label: 1. Information associated with a vertex or edge of a graph. A labeled graph is a graph whose vertices or edges have labels. The termsvertex-labeled oredge-labeled may be used to specify which objects of a graph have labels.Graph labeling refers to several different problems of assigning labels to graphs subject to certain constraints. See alsograph coloring, in which the labels are interpreted as colors.; 2. In the context ofgraph enumeration, the vertices of a graph are said to be labeled if they are all distinguishable from each other. For instance, this can be made to be true by fixing a one-to-one correspondence between the vertices and the integers from 1 to the order of the graph. When vertices are labeled, graphs that are isomorphic to each other (but with different vertex orderings) are counted as separate objects. In contrast, when the vertices are unlabeled, graphs that are isomorphic to each other are not counted separately.
leaf: 1. A leaf vertex or pendant vertex (especially in a tree) is a vertex whose degree is 1. A leaf edge or pendant edge is the edge connecting a leaf vertex to its single neighbour.; 2. Aleaf power of a tree is a graph whose vertices are the leaves of the tree and whose edges connect leaves whose distance in the tree is at most a given threshold.
length: In an unweighted graph, the length of a cycle, path, or walk is the number of edges it uses. In a weighted graph, it may instead be the sum of the weights of the edges that it uses. Length is used to define theshortest path,girth (shortest cycle length), andlongest path between two vertices in a graph.
level: 1. This is thedepth of a node plus 1, although some^[12] define it instead to be synonym ofdepth. A node's level in a rooted tree is the number of nodes in the path from the root to the node. For instance, the root has level 1 and any one of its adjacent nodes has level 2.; 2. A set of all node having the same level or depth.^[12]
line: A synonym for an undirected edge. Theline graphL(G) of a graphG is a graph with a vertex for each edge ofG and an edge for each pair of edges that share an endpoint inG.
linkage: A synonym fordegeneracy.
list: 1. Anadjacency list is a computer representation of graphs for use in graph algorithms.; 2. List coloring is a variation of graph coloring in which each vertex has a list of available colors.
local: A local property of a graph is a property that is determined only by theneighbourhoods of the vertices in the graph. For instance, a graph is locally finite if all of its neighborhoods are finite.
loop: Aloop or self-loop is an edge both of whose endpoints are the same vertex. It forms a cycle of length1. These are not allowed in simple graphs.

M

magnification: Synonym for vertexexpansion.
matching: Amatching is a set of edges in which no two share any vertex. A vertex is matched or saturated if it is one of the endpoints of an edge in the matching. Aperfect matching or complete matching is a matching that matches every vertex; it may also be called a 1-factor, and can only exist when the order is even. A near-perfect matching, in a graph with odd order, is one that saturates all but one vertex. Amaximum matching is a matching that uses as many edges as possible; the matching numberα′(G) of a graphG is the number of edges in a maximum matching. Amaximal matching is a matching to which no additional edges can be added.
maximal: 1. A subgraph of given graphG is maximal for a particular property if it has that property but no other supergraph of it that is also a subgraph ofG also has the same property. That is, it is amaximal element of the subgraphs with the property. For instance, amaximal clique is a complete subgraph that cannot be expanded to a larger complete subgraph. The word "maximal" should be distinguished from "maximum": a maximum subgraph is always maximal, but not necessarily vice versa.; 2. A simple graph with a given property is maximal for that property if it is not possible to add any more edges to it (keeping the vertex set unchanged) while preserving both the simplicity of the graph and the property. Thus, for instance, amaximal planar graph is a planar graph such that adding any more edges to it would create a non-planar graph.
maximum: A subgraph of a given graphG is maximum for a particular property if it is the largest subgraph (by order or size) among all subgraphs with that property. For instance, amaximum clique is any of the largest cliques in a given graph.
median: 1. A median of a triple of vertices, a vertex that belongs to shortest paths between all pairs of vertices, especially in median graphs andmodular graphs.; 2. Amedian graph is a graph in which every three vertices have a unique median.
Meyniel: 1. Henri Meyniel, French graph theorist.; 2. AMeyniel graph is a graph in which every odd cycle of length five or more has at least two chords.
minimal: A subgraph of given graph is minimal for a particular property if it has that property but no proper subgraph of it also has the same property. That is, it is aminimal element of the subgraphs with the property.
minimum cut: Acut whosecut-set has minimum total weight, possibly restricted to cuts that separate a designated pair of vertices; they are characterized by themax-flow min-cut theorem.
minor: A graphH is aminor of another graphG ifH can be obtained by deleting edges or vertices fromG and contracting edges inG. It is ashallow minor if it can be formed as a minor in such a way that the subgraphs ofG that were contracted to form vertices ofH all have small diameter.H is atopological minor ofG ifG has a subgraph that is asubdivision ofH. A graph isH-minor-free if it does not haveH as a minor. A family of graphs is minor-closed if it is closed under minors; theRobertson–Seymour theorem characterizes minor-closed families as having a finite set offorbidden minors.
mixed: Amixed graph is a graph that may include both directed and undirected edges.
modular: 1. Modular graph, a graph in which each triple of vertices has at least one median vertex that belongs to shortest paths between all pairs of the triple.; 2. Modular decomposition, a decomposition of a graph into subgraphs within which all vertices connect to the rest of the graph in the same way.; 3. Modularity of a graph clustering, the difference of the number of cross-cluster edges from its expected value.
monotone: A monotone property of graphs is a property that is closed under subgraphs: ifG has a monotone property, then so must every subgraph ofG. Comparehereditary (closed under induced subgraphs) orminor-closed (closed under minors).
Moore graph: AMoore graph is a regular graph for which the Moore bound is met exactly. The Moore bound is an inequality relating the degree, diameter, and order of a graph, proved byEdward F. Moore. Every Moore graph is a cage.
multigraph: Amultigraph is a graph that allows multiple adjacencies (and, often, self-loops); a graph that is not required to be simple.
multiple adjacency: A multiple adjacency or multiple edge is a set of more than one edge that all have the same endpoints (in the same direction, in the case of directed graphs). A graph with multiple edges is often called a multigraph.
multiplicity: The multiplicity of an edge is the number of edges in a multiple adjacency. The multiplicity of a graph is the maximum multiplicity of any of its edges.

N

N: 1. For the notation for open and closed neighborhoods, seeneighbourhood.; 2. A lower-casen is often used (especially in computer science) to denote the number of vertices in a given graph.
neighbor
neighbour: A vertex that is adjacent to a given vertex.
neighborhood
neighbourhood: Theopen neighbourhood (or neighborhood) of a vertexv is the subgraph induced by all vertices that are adjacent tov. The closed neighbourhood is defined in the same way but also includesv itself. The open neighborhood ofv inG may be denotedN_G(v) orN(v), and the closed neighborhood may be denotedN_G[v] orN[v]. When the openness or closedness of a neighborhood is not specified, it is assumed to be open.
network: A graph in which attributes (e.g. names) are associated with the nodes and/or edges.
node: A synonym forvertex.
non-edge: A non-edge or anti-edge is a pair of vertices that are not adjacent; the edges of the complement graph.
null graph: Seeempty graph.

O

odd: 1. An odd cycle is a cycle whose length is odd. Theodd girth of a non-bipartite graph is the length of its shortest odd cycle. An odd hole is a special case of an odd cycle: one that is induced and has four or more vertices.; 2. An odd vertex is a vertex whose degree is odd. By thehandshaking lemma every finite undirected graph has an even number of odd vertices.; 3. An odd ear is a simple path or simple cycle with an odd number of edges, used in odd ear decompositions of factor-critical graphs; seeear.; 4. An odd chord is an edge connecting two vertices that are an odd distance apart in an even cycle. Odd chords are used to definestrongly chordal graphs.; 5. Anodd graph is a special case of aKneser graph, having one vertex for each(n − 1)-element subset of a(2n − 1)-element set, and an edge connecting two subsets when their corresponding sets are disjoint.
open: 1. Seeneighbourhood.; 2. Seewalk.
order: 1. The order of a graphG is the number of its vertices,|V(G)|. The variablen is often used for this quantity. See alsosize, the number of edges.; 2. A type oflogic of graphs; seefirst order andsecond order.; 3. An order or ordering of a graph is an arrangement of its vertices into a sequence, especially in the context oftopological ordering (an order of a directed acyclic graph in which every edge goes from an earlier vertex to a later vertex in the order) anddegeneracy ordering (an order in which each vertex has minimum degree in the induced subgraph of it and all later vertices).; 4. For the order of a haven or bramble, seehaven andbramble.
orientation
oriented: 1. Anorientation of an undirected graph is an assignment of directions to its edges, making it into a directed graph. An oriented graph is one that has been assigned an orientation. So, for instance, apolytree is an oriented tree; it differs from a directed tree (an arborescence) in that there is no requirement of consistency in the directions of its edges. Other special types of orientation includetournaments, orientations of complete graphs;strong orientations, orientations that are strongly connected;acyclic orientations, orientations that are acyclic;Eulerian orientations, orientations that are Eulerian; andtransitive orientations, orientations that are transitively closed.; 2. Oriented graph, used by some authors as a synonym for adirected graph.
out-degree: Seedegree.
outer: Seeface.
outerplanar: Anouterplanar graph is a graph that can be embedded in the plane (without crossings) so that all vertices are on the outer face of the graph.

P

parent: In a rooted tree, a parent of a vertexv is a neighbor ofv along the incoming edge, the one that is directed toward the root.
path: Apath may either be a walk or a walk without repeated vertices and consequently edges (also called a simple path), depending on the source. Important special cases includeinduced paths andshortest paths.
path decomposition: Apath decomposition of a graphG is a tree decomposition whose underlying tree is a path. Its width is defined in the same way as for tree decompositions, as one less than the size of the largest bag. The minimum width of any path decomposition ofG is the pathwidth ofG.
pathwidth: Thepathwidth of a graphG is the minimum width of a path decomposition ofG. It may also be defined in terms of the clique number of an interval completion ofG. It is always between the bandwidth and the treewidth ofG. It is also known as interval thickness, vertex separation number, or node searching number.
pendant: Seeleaf.
perfect: 1. Aperfect graph is a graph in which, in every induced subgraph, the chromatic number equals the clique number. Theperfect graph theorem andstrong perfect graph theorem are two theorems about perfect graphs, the former proving that their complements are also perfect and the latter proving that they are exactly the graphs with no odd holes or anti-holes.; 2. Aperfectly orderable graph is a graph whose vertices can be ordered in such a way that a greedy coloring algorithm with this ordering optimally colors every induced subgraph. The perfectly orderable graphs are a subclass of the perfect graphs.; 3. Aperfect matching is a matching that saturates every vertex; seematching.; 4. A perfect1-factorization is a partition of the edges of a graph into perfect matchings so that each two matchings form a Hamiltonian cycle.
peripheral: 1. Aperipheral cycle or non-separating cycle is a cycle with at most one bridge.; 2. A peripheral vertex is a vertex whoseeccentricity is maximum. In a tree, this must be a leaf.
Petersen: 1. Julius Petersen (1839–1910), Danish graph theorist.; 2. ThePetersen graph, a 10-vertex 15-edge graph frequently used as a counterexample.; 3. Petersen's theorem that every bridgeless cubic graph has a perfect matching.
planar: Aplanar graph is a graph that has anembedding onto the Euclidean plane. A plane graph is a planar graph for which a particular embedding has already been fixed. Ak-planar graph is one that can be drawn in the plane with at mostk crossings per edge.
polytree: Apolytree is an oriented tree; equivalently, a directed acyclic graph whose underlying undirected graph is a tree.
power: 1. Agraph powerG^k of a graphG is another graph on the same vertex set; two vertices are adjacent inG^k when they are at distance at mostk inG. Aleaf power is a closely related concept, derived from a power of a tree by taking the subgraph induced by the tree's leaves.; 2. Power graph analysis is a method for analyzing complex networks by identifying cliques, bicliques, and stars within the network.; 3. Power laws in thedegree distributions ofscale-free networks are a phenomenon in which the number of vertices of a given degree is proportional to a power of the degree.
predecessor: Avertex coming before a given vertex in adirected path.
prime: 1. Aprime graph is defined from an algebraicgroup, with a vertex for eachprime number that divides the order of the group.; 2. In the theory ofmodular decomposition, a prime graph is a graph without any nontrivial modules.; 3. In the theory ofsplits, cuts whose cut-set is a complete bipartite graph, a prime graph is a graph without any splits. Every quotient graph of a maximal decomposition by splits is a prime graph, a star, or a complete graph.; 4. A prime graph for theCartesian product of graphs is a connected graph that is not itself a product. Every connected graph can be uniquely factored into a Cartesian product of prime graphs.
proper: 1. A proper subgraph is a subgraph that removes at least one vertex or edge relative to the whole graph; for finite graphs, proper subgraphs are never isomorphic to the whole graph, but for infinite graphs they can be.; 2. A proper coloring is an assignment of colors to the vertices of a graph (a coloring) that assigns different colors to the endpoints of each edge; seecolor.; 3. Aproper interval graph or proper circular arc graph is an intersection graph of a collection of intervals or circular arcs (respectively) such that no interval or arc contains another interval or arc. Proper interval graphs are also called unit interval graphs (because they can always be represented by unit intervals) or indifference graphs.
property: Agraph property is something that can be true of some graphs and false of others, and that depends only on the graph structure and not on incidental information such as labels. Graph properties may equivalently be described in terms of classes of graphs (the graphs that have a given property). More generally, a graph property may also be a function of graphs that is again independent of incidental information, such as the size, order, or degree sequence of a graph; this more general definition of a property is also called an invariant of the graph.
pseudoforest: Apseudoforest is an undirected graph in which each connected component has at most one cycle, or a directed graph in which each vertex has at most one outgoing edge.
pseudograph: A pseudograph is a graph or multigraph that allows self-loops.

Q

quasi-line graph: A quasi-line graph or locally co-bipartite graph is a graph in which the open neighborhood of every vertex can be partitioned into two cliques. These graphs are alwaysclaw-free and they include as a special case theline graphs. They are used in the structure theory of claw-free graphs.
quasi-random graph sequence: Aquasi-random graph sequence is a sequence of graphs that shares several properties with a sequence ofrandom graphs generated according to theErdős–Rényi random graph model.
quiver: Aquiver is a directed multigraph, as used incategory theory. The edges of a quiver are called arrows.

R

radius: The radius of a graph is the minimumeccentricity of any vertex.
Ramanujan: ARamanujan graph is a graph whose spectral expansion is as large as possible. That is, it is ad-regular graph, such that the second-largest eigenvalue of its adjacency matrix is at most $2{\sqrt {d-1}}$ .
ray: A ray, in an infinite graph, is an infinite simple path with exactly one endpoint. Theends of a graph are equivalence classes of rays.
reachability: The ability to get from onevertex to another within agraph.
reachable: Has an affirmativereachability. Avertexy is said to be reachable from a vertexx if there exists apath fromx toy.
recognizable: In the context of thereconstruction conjecture, a graph property is recognizable if its truth can be determined from the deck of the graph. Many graph properties are known to be recognizable. If the reconstruction conjecture is true, all graph properties are recognizable.
reconstruction: Thereconstruction conjecture states that each undirected graphG is uniquely determined by itsdeck, a multiset of graphs formed by removing one vertex fromG in all possible ways. In this context, reconstruction is the formation of a graph from its deck.
rectangle: A simple cycle consisting of exactly four edges and four vertices.
regular: A graph isd-regular when all of its vertices have degreed. Aregular graph is a graph that isd-regular for somed.
regular tournament: A regular tournament is a tournament where in-degree equals out-degree for all vertices.
reverse: Seetranspose.
root: 1. A designated vertex in a graph, particularly in directed trees androoted graphs.; 2. The inverse operation to agraph power: akth root of a graphG is another graph on the same vertex set such that two vertices are adjacent inG if and only if they have distance at mostk in the root.

S

saturated: Seematching.
searching number: Node searching number is a synonym forpathwidth.
second order: The second orderlogic of graphs is a form of logic in which variables may represent vertices, edges, sets of vertices, and (sometimes) sets of edges. This logic includes predicates for testing whether a vertex and edge are incident, as well as whether a vertex or edge belongs to a set. To be distinguished from first order logic, in which variables can only represent vertices.
self-loop: Synonym forloop.
separating vertex: Seearticulation point.
separation number: Vertex separation number is a synonym forpathwidth.
sibling: In a rooted tree, a sibling of a vertexv is a vertex which has the same parent vertex asv.
simple: 1. Asimple graph is a graph without loops and without multiple adjacencies. That is, each edge connects two distinct endpoints and no two edges have the same endpoints. A simple edge is an edge that is not part of a multiple adjacency. In many cases, graphs are assumed to be simple unless specified otherwise.; 2. A simple path or a simple cycle is a path or cycle that has no repeated vertices and consequently no repeated edges.
sink: A sink, in a directed graph, is a vertex with no outgoing edges (out-degree equals 0).
size: The size of a graphG is the number of its edges,|E(G)|.^[13] The variablem is often used for this quantity. See alsoorder, the number of vertices.
small-world network: Asmall-world network is a graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other node by a small number of hops or steps. Specifically, a small-world network is defined to be a graph where the typical distanceL between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodesN in the network^[14]
snark: Asnark is a simple, connected, bridgeless cubic graph with chromatic index equal to 4.
source: A source, in a directed graph, is a vertex with no incoming edges (in-degree equals 0).
space: Inalgebraic graph theory, severalvector spaces over thebinary field may be associated with a graph. Each has sets of edges or vertices for its vectors, andsymmetric difference of sets as its vector sum operation. Theedge space is the space of all sets of edges, and thevertex space is the space of all sets of vertices. Thecut space is a subspace of the edge space that has the cut-sets of the graph as its elements. Thecycle space has the Eulerian spanning subgraphs as its elements.
spanner: A spanner is a (usually sparse) graph whose shortest path distances approximate those in a dense graph or other metric space. Variations includegeometric spanners, graphs whose vertices are points in a geometric space;tree spanners, spanning trees of a graph whose distances approximate the graph distances, and graph spanners, sparse subgraphs of a dense graph whose distances approximate the original graph's distances. A greedy spanner is a graph spanner constructed by a greedy algorithm, generally one that considers all edges from shortest to longest and keeps the ones that are needed to preserve the distance approximation.
spanning: A subgraph is spanning when it includes all of the vertices of the given graph.Important cases includespanning trees, spanning subgraphs that are trees, andperfect matchings, spanning subgraphs that are matchings. A spanning subgraph may also be called afactor, especially (but not only) when it is regular.
sparse: Asparse graph is one that has few edges relative to its number of vertices. In some definitions the same property should also be true for all subgraphs of the given graph.
spectral
spectrum: The spectrum of a graph is the collection ofeigenvalues of its adjacency matrix.Spectral graph theory is the branch of graph theory that uses spectra to analyze graphs. See also spectralexpansion.
split: 1. Asplit graph is a graph whose vertices can be partitioned into a clique and an independent set. A related class of graphs, the double split graphs, are used in the proof of the strong perfect graph theorem.; 2. Asplit of an arbitrary graph is a partition of its vertices into two nonempty subsets, such that the edges spanning this cut form a complete bipartite subgraph. The splits of a graph can be represented by a tree structure called itssplit decomposition. A split is called a strong split when it is not crossed by any other split. A split is called nontrivial when both of its sides have more than one vertex. A graph is called prime when it has no nontrivial splits.; 3. Vertex splitting (sometimes called vertex cleaving) is an elementary graph operation that splits a vertex into two, where these two new vertices are adjacent to the vertices that the original vertex was adjacent to. The inverse of vertex splitting is vertex contraction.
square: 1. The square of a graphG is thegraph powerG²; in the other direction,G is the square root ofG². Thehalf-square of a bipartite graph is the subgraph of its square induced by one side of the bipartition.; 2. Asquaregraph is a planar graph that can be drawn so that all bounded faces are 4-cycles and all vertices of degree ≤ 3 belong to the outer face.; 3. A square grid graph is alattice graph defined from points in the plane with integer coordinates connected by unit-length edges.
stable: A stable set is a synonym for anindependent set.
star: Astar is a tree with one internal vertex; equivalently, it is a complete bipartite graphK_1,n for somen ≥ 2. The special case of a star with three leaves is called a claw.
strength: Thestrength of a graph is the minimum ratio of the number of edges removed from the graph to components created, over all possible removals; it is analogous to toughness, based on vertex removals.
strong: 1. For strong connectivity andstrongly connected components of directed graphs, seeconnected andcomponent. Astrong orientation is an orientation that is strongly connected; seeorientation.; 2. For thestrong perfect graph theorem, seeperfect.; 3. Astrongly regular graph is a regular graph in which every two adjacent vertices have the same number of shared neighbours and every two non-adjacent vertices have the same number of shared neighbours.; 4. Astrongly chordal graph is a chordal graph in which every even cycle of length six or more has an odd chord.; 5. A strongly perfect graph is a graph in which every induced subgraph has an independent set meeting all maximal cliques. TheMeyniel graphs are also called "very strongly perfect graphs" because in them, every vertex belongs to such an independent set.
subforest: A subgraph of aforest.
subgraph: A subgraph of a graphG is another graph formed from a subset of the vertices and edges ofG. The vertex subset must include all endpoints of the edge subset, but may also include additional vertices. A spanning subgraph is one that includes all vertices of the graph; an induced subgraph is one that includes all the edges whose endpoints belong to the vertex subset.
subtree: A subtree is a connected subgraph of a tree. Sometimes, for rooted trees, subtrees are defined to be a special type of connected subgraph, formed by all vertices and edges reachable from a chosen vertex.
successor: Avertex coming after a given vertex in adirected path.
superconcentrator: A superconcentrator is a graph with two designated and equal-sized subsets of verticesI andO, such that for every two equal-sized subsetsS ofI andT ofO there exists a family of disjoint paths connecting every vertex inS to a vertex inT. Some sources require in addition that a superconcentrator be a directed acyclic graph, withI as its sources andO as its sinks.
supergraph: A graph formed by adding vertices, edges, or both to a given graph. IfH is a subgraph ofG, thenG is a supergraph ofH.

T

theta: 1. A theta graph is the union of three internally disjoint (simple) paths that have the same two distinct end vertices.^[15]; 2. Thetheta graph of a collection of points in the Euclidean plane is constructed by constructing a system of cones surrounding each point and adding one edge per cone, to the point whose projection onto a central ray of the cone is smallest.; 3. TheLovász number or Lovász theta function of a graph is a graph invariant related to the clique number and chromatic number that can be computed in polynomial time by semidefinite programming.
Thomsen graph: TheThomsen graph is a name for thecomplete bipartite graph $K_{3,3}$ .
topological: 1. Atopological graph is a representation of the vertices and edges of a graph by points and curves in the plane (not necessarily avoiding crossings).; 2. Topological graph theory is the study of graph embeddings.; 3. Topological sorting is the algorithmic problem of arranging a directed acyclic graph into a topological order, a vertex sequence such that each edge goes from an earlier vertex to a later vertex in the sequence.
totally disconnected: Synonym foredgeless.
tour: A closed trail, awalk that starts and ends at the same vertex and has no repeated edges. Euler tours are tours that use all of the graph edges; seeEulerian.
tournament: Atournament is an orientation of a complete graph; that is, it is a directed graph such that every two vertices are connected by exactly one directed edge (going in only one of the two directions between the two vertices).
traceable: Atraceable graph is a graph that contains a Hamiltonian path.
trail: Awalk without repeated edges.
transitive: Having to do with thetransitive property. Thetransitive closure of a given directed graph is a graph on the same vertex set that has an edge from one vertex to another whenever the original graph has a path connecting the same two vertices. Atransitive reduction of a graph is a minimal graph having the same transitive closure; directed acyclic graphs have a unique transitive reduction. Atransitive orientation is an orientation of a graph that is its own transitive closure; it exists only forcomparability graphs.
transpose: Thetranspose graph of a given directed graph is a graph on the same vertices, with each edge reversed in direction. It may also be called the converse or reverse of the graph.
tree: 1. Atree is an undirected graph that is both connected and acyclic, or a directed graph in which there exists a unique walk from one vertex (the root of the tree) to all remaining vertices.; 2. Ak-tree is a graph formed by gluing(k + 1)-cliques together on sharedk-cliques. A tree in the ordinary sense is a1-tree according to this definition.
tree decomposition: Atree decomposition of a graphG is a tree whose nodes are labeled with sets of vertices ofG; these sets are called bags. For each vertexv, the bags that containv must induce a subtree of the tree, and for each edgeuv there must exist a bag that contains bothu andv. The width of a tree decomposition is one less than the maximum number of vertices in any of its bags; the treewidth ofG is the minimum width of any tree decomposition ofG.
treewidth: Thetreewidth of a graphG is the minimum width of a tree decomposition ofG. It can also be defined in terms of the clique number of achordal completion ofG, the order of ahaven ofG, or the order of abramble ofG.
triangle: A cycle of length three in a graph. Atriangle-free graph is an undirected graph that does not have any triangle subgraphs.
trivial: A trivial graph is a graph with 0 or 1 vertices.^[16] A graph with 0 vertices is also callednull graph.
Turán: 1. Pál Turán; 2. ATurán graph is a balanced complete multipartite graph.; 3. Turán's theorem states that Turán graphs have the maximum number of edges among all clique-free graphs of a given order.; 4. Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph.
twin: Two verticesu,v are true twins if they have the same closedneighborhood:N_G[u] =N_G[v] (this impliesu andv are neighbors), and they are false twins if they have the same open neighborhood:N_G(u) =N_G(v)) (this impliesu andv are not neighbors).

U

unary vertex: In a rooted tree, a unary vertex is a vertex which has exactly one child vertex.
undirected: Anundirected graph is a graph in which the two endpoints of each edge are not distinguished from each other. See alsodirected andmixed. In amixed graph, an undirected edge is again one in which the endpoints are not distinguished from each other.
uniform: A hypergraph isk-uniform when all its edges havek endpoints, and uniform when it isk-uniform for somek. For instance, ordinary graphs are the same as2-uniform hypergraphs.
universal: 1. Auniversal graph is a graph that contains as subgraphs all graphs in a given family of graphs, or all graphs of a given size or order within a given family of graphs.; 2. Auniversal vertex (also called an apex or dominating vertex) is a vertex that is adjacent to every other vertex in the graph. For instance,wheel graphs and connectedthreshold graphs always have a universal vertex.; 3. In thelogic of graphs, a vertex that isuniversally quantified in a formula may be called a universal vertex for that formula.
unweighted graph: Agraph whosevertices andedges have not been assignedweights; the opposite of aweighted graph.
utility graph: Theutility graph is a name for thecomplete bipartite graph $K_{3,3}$ .

V

V: Seevertex set.
valency: Synonym fordegree.
vertex: Avertex (plural vertices) is (together with edges) one of the two basic units out of which graphs are constructed. Vertices of graphs are often considered to be atomic objects, with no internal structure.
vertex cut
separating set: A set ofvertices whose removaldisconnects thegraph. A one-vertex cut is called anarticulation point orcut vertex.
vertex set: The set of vertices of a given graphG, sometimes denoted byV(G).
vertices: Seevertex.
Vizing: 1. Vadim G. Vizing; 2. Vizing's theorem that the chromatic index is at most one more than the maximum degree.; 3. Vizing's conjecture on the domination number of Cartesian products of graphs.
volume: The sum of the degrees of a set of vertices.

W

W: The letterW is used in notation forwheel graphs andwindmill graphs. The notation is not standardized.
Wagner: 1. Klaus Wagner; 2. TheWagner graph, an eight-vertex Möbius ladder.; 3. Wagner's theorem characterizing planar graphs by their forbidden minors.; 4. Wagner's theorem characterizing theK₅-minor-free graphs.
walk: Awalk is a finite or infinitesequence ofedges which joins a sequence ofvertices. Walks are also sometimes calledchains.^[17] A walk isopen if its first and last vertices are distinct, andclosed if they are repeated.
weakly connected: Adirected graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph.
weight: A numerical value, assigned as a label to a vertex or edge of a graph. The weight of a subgraph is the sum of the weights of the vertices or edges within that subgraph.
weighted graph: Agraph whosevertices oredges have been assignedweights. A vertex-weighted graph has weights on its vertices and an edge-weighted graph has weights on its edges.
well-colored: Awell-colored graph is a graph all of whosegreedy colorings use the same number of colors.
well-covered: Awell-covered graph is a graph all of whose maximal independent sets are the same size.
wheel: Awheel graph is a graph formed by adding auniversal vertex to a simple cycle.
width: 1. A synonym fordegeneracy.; 2. For other graph invariants known as width, seebandwidth,branchwidth,clique-width,pathwidth, andtreewidth.; 3. The width of a tree decomposition or path decomposition is one less than the maximum size of one of its bags, and may be used to define treewidth and pathwidth.; 4. The width of adirected acyclic graph is the maximum cardinality of an antichain.
windmill: Awindmill graph is the union of a collection of cliques, all of the same order as each other, with one shared vertex belonging to all the cliques and all other vertices and edges distinct.

See also

References

^Farber, M.; Hahn, G.;Hell, P.; Miller, D. J. (1986), "Concerning the achromatic number of graphs",Journal of Combinatorial Theory, Series B,40 (1):21–39,doi:10.1016/0095-8956(86)90062-6.
^^a ^b ^c ^d ^e ^f ^g ^hCormen, Thomas H.;Leiserson, Charles E.;Rivest, Ronald L.;Stein, Clifford (2001), "B.4 Graphs",Introduction to Algorithms (2 ed.), MIT Press and McGraw-Hill, pp. 1080–1084.
^Grünbaum, B. (1973), "Acyclic colorings of planar graphs",Israel Journal of Mathematics,14 (4):390–408,doi:10.1007/BF02764716.
^Cormen et al. (2001), p. 529.
^Diestel, Reinhard (2017), "1.1 Graphs",Graph Theory, Graduate Texts in Mathematics, vol. 173 (5th ed.), Berlin, New York: Springer-Verlag, p. 3,doi:10.1007/978-3-662-53622-3,ISBN 978-3-662-53621-6.
^Woodall, D. R. (1973), "The Binding Number of a Graph and its Anderson Number",J. Combin. Theory Ser. B,15 (3):225–255,doi:10.1016/0095-8956(73)90038-5
^van der Holst, Hein (March 2009), "A polynomial-time algorithm to find a linkless embedding of a graph",Journal of Combinatorial Theory, Series B,99 (2), Elsevier BV:512–530,doi:10.1016/j.jctb.2008.10.002
^Sudakov, Benny; Volec, Jan (2017), "Properly colored and rainbow copies of graphs with few cherries",Journal of Combinatorial Theory, Series B,122 (1):391–416,arXiv:1504.06176,doi:10.1016/j.jctb.2016.07.001.
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^Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), "Chapter 7: Forbidden Subgraph",Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, pp. 105–121,ISBN 978-0-89871-432-6
^Mitchem, John (1969), "Hypo-properties in graphs",The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968), Lecture Notes in Mathematics, vol. 110, Springer, pp. 223–230,doi:10.1007/BFb0060121,ISBN 978-3-540-04629-5,MR 0253932.
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^Harris, John M. (2000),Combinatorics and Graph Theory, New York: Springer-Verlag, p. 5,ISBN 978-0-387-98736-1
^Watts, Duncan J.; Strogatz, Steven H. (June 1998), "Collective dynamics of 'small-world' networks",Nature,393 (6684):440–442,Bibcode:1998Natur.393..440W,doi:10.1038/30918,PMID 9623998,S2CID 4429113
^Bondy, J. A. (1972), "The "graph theory" of the Greek alphabet",Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1972; dedicated to the memory of J. W. T. Youngs), Lecture Notes in Mathematics, vol. 303, Springer, pp. 43–54,doi:10.1007/BFb0067356,ISBN 978-3-540-06096-3,MR 0335362
^Diestel, Reinhard (2017),Graph Theory, Graduate Texts in Mathematics, vol. 173, Berlin, Heidelberg: Springer Berlin Heidelberg, p. 2,doi:10.1007/978-3-662-53622-3,ISBN 978-3-662-53621-6
^"Chain - graph theory",britannica.com, retrieved25 March 2018

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