Clique or hole in claw-free graphs

Authors:
Henning Bruhn;Akira Saito
Affiliations:
íquipe Combinatoire et Optimisation, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France;Department of Computer Science and System Analysis, Nihon University, Sakurajosui 3-25-40, Setagaya-Ku Tokyo 156-8550, Japan
Venue:
Journal of Combinatorial Theory Series B
Year:
2012

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Abstract

Given a claw-free graph and two non-adjacent vertices x and y without common neighbours we prove that there exists a hole through x and y unless the graph contains the obvious obstruction, namely a clique separating x and y. We derive two applications: We give a necessary and sufficient condition for the existence of an induced x-z path through y, where x,y,z are prescribed vertices in a claw-free graph; and we prove an induced version of Menger@?s theorem between four terminal vertices. Finally, we improve the running time for detecting a hole through x and y and for the Three-in-a-Tree problem, if the input graph is claw-free.