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

For all integers k 3, we give an O(n4 ) 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). AMS Mathematics Subject Classification: 05C75, 05C85, 05C05, 68R10, 90C35 Key words: tree, algorithm, three-in-a-tree, k-in-a-tree, girth, induced subgraph.

• Type: Journal
• Year: 2010
• Where: DAM
• Authors: W. Liu, Nicolas Trotignon
• Keywords: Algorithm, DAM 2010, Three-in-a-tree Algorithm, Time Algorithm