The Lexicographic Method for the Threshold Cover Problem
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The lexicographic method is a technique that was introduced by Hell and Huang [Journal of Graph Theory, 20(3):361–374, 1995] as a way to simplify the problems of recognizing and obtaining representations of comparability graphs, proper circular-arc graphs and proper interval graphs. This method gives rise to conceptually simple recognition algorithms and leads to much simpler proofs for some characterization theorems for these classes. Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size k if its edges can be covered using k threshold graphs. Chvátal and Hammer conjectured in 1977 that given a graph G, a suitably constructed auxiliary graph G' has chromatic number equal to the minimum size of a threshold cover of G. Although Cozzens and Leibowitz showed that this conjecture is false in the general case, Raschle and Simon [Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 650–661, 1995] proved that G has a threshold cover of size 2 if and only if G' is bipartite. We show how the lexicographic method can be used to obtain a completely new and much simpler proof for this result. This method also gives rise to a simple new LexBFS-based algorithm for recognizing graphs having a threshold cover of size 2. Although this algorithm is not the fastest known, it is a certifying algorithm that matches the time complexity of the fastest known certifying algorithm for this problem. The algorithm can also be easily adapted to give a certifying recognition algorithm for bipartite graphs that can be covered by two chain subgraphs.
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