A new characterization of P6-free graphs

  • Authors:

  • Pim van 't Hof;Daniël Paulusma

  • Affiliations:

  • Department of Computer Science, Durham University, Science Laboratories, South Road, Durham DH1 3LE, England, United Kingdom;Department of Computer Science, Durham University, Science Laboratories, South Road, Durham DH1 3LE, England, United Kingdom

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

We study P"6-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 G is P"6-free if and only if each connected induced subgraph of G on 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 P"6-free graphs for which a smallest connected dominating subgraph is a (not induced) complete bipartite graph. Our characterization of P"6-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 P"6-free graph in polynomial time. This enables us to solve the Hypergraph 2-Colorability problem in polynomial time for the class of hypergraphs with P"6-free incidence graphs.