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The jump number problem for biconvex graphs and rectangle covers of rectangular regions

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Part of the book series:Lecture Notes in Computer Science ((LNCS,volume 380))

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Abstract

Let P=(V, ≤p) be a finite partially ordered set (poset) with |V|=n and let L=(1l,...,1n) be a linear extension of P. The pair (1i,1i+1), 1≤i≤n-1, is a jump of P in L if

. The jump number problem is the problem of finding the minimum number of jumps in any linear extension of a given poset P. It is known that for posets P1,P2 with the same comparability graph also the jump numbers of P1 and P2 coincide and that for chordal bipartite graphs the jump number decision problem is NP-complete.

We show in this paper that the jump number of biconvex graphs (a subclass of chordal bipartite graphs) can be determined in polynomial time using several reformulations of the problem and a duality relation between rectangle independent point sets and rectangle covers of rectangular regions known from a result of Chaiken/Kleitman/Saks/Shearer and Franzblau/Kleitman. This solves the jump number problem for biconvex graphs by means of computational geometry. Furthermore for bipartite permutation graphs (a subclass of biconvex graphs) the rectangle cover approach yields a greedy solution which is faster than the dynamic programming solution given by Steiner/Stewart. An optimal rectangle cover can be determined in linear time using the geometric description of the region or the defining permutation.

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Authors and Affiliations

  1. Sektion Mathematik, Friedrich-Schiller-Universität Jena, Universitätshochhaus, DDR-6900, Jena

    Andreas Brandstädt

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  1. Andreas Brandstädt

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J. Csirik J. Demetrovics F. Gécseg

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© 1989 Springer-Verlag Berlin Heidelberg

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Brandstädt, A. (1989). The jump number problem for biconvex graphs and rectangle covers of rectangular regions. In: Csirik, J., Demetrovics, J., Gécseg, F. (eds) Fundamentals of Computation Theory. FCT 1989. Lecture Notes in Computer Science, vol 380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51498-8_7

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