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

09/05/2013

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by   Wei Liu, et al.

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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).