Finding Bipartite Partitions on Co-Chordal Graphs

Abstract

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.

Maximal Clique and Edge-Ranking Bounds of Biclique Cover Number

“…return {{L, R}} ∪ FindPartition(T K c 1 ) ∪ FindPartition(T K c 2 ) 10: end function Theorem 4.1 (Theorems 1 and 2 [21]). Given a co-chordal graph G and clique tree T K c of its complement G c , the output of FindPartition(T K c ) is a biclique partition of G with size mc(G c ) − 1.…”