Finding k disjoint triangles in an arbitrary graph

Abstract

We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. For H = K3 (the triangle) we give an O(2 2k log k+1.869k n2) algorithm, and for general H an O(2k|H| log k+2k|H| log |H|n|H|) algorithm. We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph. For H = K3 this leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k3) vertices in polynomial time. © Springer-Verlag 2004.