The k-in-a-tree problem for graphs of girth at least k

Authors:
W. Liu;N. Trotignon
Affiliations:
Université Grenoble 1, Joseph Fourier, France;CNRS, LIAFA, Université Paris 7, Paris Diderot, France
Venue:
Discrete Applied Mathematics
Year:
2010

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Abstract

For all integers k=3, we give an O(n^4)-time algorithm for the problem whose instance is a graph G of girth at least k together with k vertices and whose question is ''Does G contains an induced subgraph containing the k vertices and isomorphic to a tree?''. This directly follows for k=3 from the three-in-a-tree algorithm of Chudnovsky and Seymour and for k=4 from a result of Derhy, Picouleau and Trotignon. Here we solve the problem for k=5. Our algorithm relies on a structural description of graphs of girth at least k that do not contain an induced tree covering k given vertices (k=5).