A New Characterization of P6-Free Graphs

Authors:
Pim Hof;Daniël Paulusma
Affiliations:
Department of Computer Science, Durham University, Science Laboratories, Durham, England DH1 3LE;Department of Computer Science, Durham University, Science Laboratories, Durham, England DH1 3LE
Venue:
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Year:
2008

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Abstract

We study P6-free graphs, i.e., graphs that do not contain an induced path on six vertices. Our main result is a new characterization of this graph class: a graph Gis P6-free if and only if each connected induced subgraph of Gon more than one vertex contains a dominating induced cycle on six vertices or a dominating (not necessarily induced) complete bipartite subgraph. This characterization is minimal in the sense that there exists an infinite family of P6-free graphs for which a smallest connected dominating subgraph is a (not induced) complete bipartite graph. Our characterization of P6-free graphs strengthens results of Liu and Zhou, and of Liu, Peng and Zhao. Our proof has the extra advantage of being constructive: we present an algorithm that finds such a dominating subgraph of a connected P6-free graph in polynomial time. This enables us to solve the Hypergraph 2-Colorabilityproblem in polynomial time for the class of hypergraphs with P6-free incidence graphs.