Packing seagulls

Abstract

A seagull in a graph is an induced three-vertex path. When does a graph G have k pairwise vertex-disjoint seagulls? This is NP-complete in general, but for graphs with no stable set of size three we give a complete solution. This case is of special interest because of a connection with Hadwiger’s conjecture which was the motivation for this research; and we deduce a unification and strengthening of two theorems of Blasiak [2] concerned with Hadwiger’s conjecture. Our main result is that a graph G (different from the five-wheel) with no three-vertex stable set contains k disjoint seagulls if and only if vertical bar V vertical bar(G) >= 3K G is k-connected for every clique C of G, if D denotes the set of vertices in V (G)\textbackslashC that have both a neighbour and a non-neighbour in C then |D|+|V (G)\textbackslashC|a parts per thousand yen2k, and the complement graph of G has a matching with k edges. We also address the analogous fractional and half-integral packing questions, and give a polynomial time algorithm to test whether there are k disjoint seagulls.