Geometric intersection graph: do short cycles help?

Authors:
Jan Kratochvíl;Martin Pergel
Affiliations:
Department of Applied Mathematics, Charles University, Malostranské nám. 25, Czech Republic;Department of Applied Mathematics, Charles University, Malostranské nám. 25, Czech Republic
Venue:
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Year:
2007

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Abstract

Geometric intersection graphs are intensively studied both for their practical motivation and interesting theoretical properties. Many such classes are hard to recognize. We ask the question if imposing restrictions on the girth (the length of a shortest cycle) of the input graphs may help in finding polynomial time recognition algorithms. We give examples in both directions. First we present a polynomial time recognition algorithm for intersection graphs of polygons inscribed in a circle for inputs of girth greater than four (the general recognition problem is NP-complete). On the other hand, we prove that recognition of intersection graphs of segments in the plane remains NP-hard for graphs with arbitrarily large girth.