2022/03/06 by Bochuan Lyu, Lyu, Bochuan, Illya V. Hicks +1 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2203.02837
openalex publication_date 2022/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The biclique partition number (bp) of a graph G 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) 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 Gc = (V, Ec). 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|)] is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with size mc(Gc)-1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement Gc is chordal and clique vertex irreducible. We also show that mc(Gc) - 2 ≤ bp(G) ≤ mc(Gc) - 1 if G is a split graph.