Finding Bipartite Partitions on Co-Chordal Graphs
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In this paper, we show that the biclique partition number (bp) of a co-chordal (complementary graph of chordal) graph G = (V, E) is less than the number of maximal cliques (mc) of its complementary graph: a chordal graph G^c = (V, E^c). We first provide a general framework of the "divided and conquer" heuristic of finding minimum biclique partition on co-chordal graphs based on clique trees. Then, an O[|V|(|V|+|E^c|)]-time heuristic is proposed by applying lexicographic breadth-first search. Either heuristic gives us a biclique partition of G with a size of mc(G^c)-1. Eventually, we prove that our heuristic can solve the minimum biclique partition problem on G exactly if its complement G^c is chordal and clique vertex irreducible. We also show that mc(G^c) - 2 ≤bp(G) ≤mc(G^c) - 1 if G is a split graph.
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