Finding Biclique Partitions of Co-Chordal Graphs

TL;DR

This paper presents new heuristics and theoretical bounds for finding the minimum biclique partition number of co-chordal graphs, leveraging properties of their complements and clique structures.

Contribution

It introduces a divide and conquer heuristic based on clique trees and a lexicographic BFS-based heuristic for co-chordal graphs, with proven optimality under certain conditions.

Findings

Heuristics produce biclique partitions with size mc(G^c)-1.

Exact solutions are possible if G^c is chordal and clique vertex irreducible.

Bounds for split graphs: mc(G^c)-2 ≤ bp(G) ≤ mc(G^c)-1.

Abstract

The biclique partition number $(\text{bp})$ 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 ($\text{bp}$) of a co-chordal (complementary graph of chordal) graph $G = (V, E)$ is less than the number of maximal cliques ($\text{mc}$) of its complementary graph: a chordal graph $G^c = (V, E^c)$. 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 $O[|V|(|V|+|E^c|)]$ is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of $G$ with size $\text{mc}(G^c)-1$. In addition, we prove that both of our heuristics 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 $\text{mc}(G^c) - 2 \leq \text{bp}(G) \leq \text{mc}(G^c) - 1$ if $G$ is a split graph.