Disjoint clique cutsets in graphs without long holes
Elaine M. Eschen, Chı́nh T. Hoàng, Mark Petrick, R. Sritharan · Journal of Graph Theory · 2005
Abstract A biclique cutset is a cutset that induces the disjoint union of two cliques. A hole is an induced cycle with at least five vertices. A graph is biclique separable if it has no holes and each induced subgraph that is not a clique contains a clique cutset or a biclique cutset. The class of biclique separable graphs contains several well‐studied graph classes, including triangulated graphs. We prove that for the class of biclique separable graphs, the recognition problem, the vertex coloring problem, and the clique problem can be solved efficiently. Our algorithms also yield a proof that biclique separable graphs are perfect. Our coloring algorithm is actually more general and can be applied to graphs that can be decomposed via a special type of biclique cutset. Our algorithms are based on structural results on biclique separable graphs developed in this paper. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 277–298, 2005