Finding Biclique Partitions of Co-Chordal Graphs

Abstract

The biclique partition number (bp)(\text{bp}) of a graph GG is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number (bp\text{bp}) of a co-chordal (complementary graph of chordal) graph G=(V,E)G = (V, E) is less than the number of maximal cliques (mc\text{mc}) of its complementary graph: a chordal graph Gc=(V,Ec)G^c = (V, E^c). We first provide a general framework of the ``divide and conquer" heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O[∣V∣(∣V∣+∣Ec∣)]O[|V|(|V|+|E^c|)] is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of GG with size mc(Gc)βˆ’1\text{mc}(G^c)-1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on GG exactly if its complement GcG^c is chordal and clique vertex irreducible. We also show that mc(Gc)βˆ’2≀bp(G)≀mc(Gc)βˆ’1\text{mc}(G^c) - 2 \leq \text{bp}(G) \leq \text{mc}(G^c) - 1 if GG is a split graph

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