Cliques in odd-minor-free graphs

Authors:
Ken-ichi Kawarabayashi;David R. Wood
Affiliations:
National Institute of Informatics, Tokyo, Japan;The University of Melbourne, Melbourne, Australia
Venue:
CATS '12 Proceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 128
Year:
2012

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Abstract

This paper is about: (1) bounds on the number of cliques in a graph in a particular class, and (2) algorithms for listing all cliques in a graph. We present a simple algorithm that lists all cliques in an n-vertex graph in O(n) time per clique. For O(1)-degenerate graphs, such as graphs excluding a fixed minor, we describe a O(n) time algorithm for listing all cliques. We prove that graphs excluding a fixed odd-minor have O(n2) cliques (which is tight), and conclude a O(n3) time algorithm for listing all cliques.