The Vertex-Disjoint Triangles Problem

The vertex-disjoint triangles (VDT) problem asks for a set of maximum number of pairwise vertex-disjoint triangles in a given graph G. The triangle cover problem asks for the existence of a perfect triangle packing in a graph G. It is known that the triangle cover problem is NPcomplete on general graphs with clique number 3 [6]. The VDT problem is MAX SNP-hard on graphs with maximum degree four, while it can be approximated within 3/2 + , for any > 0, in polynomial time [11]. We prove that the VDT problem is NP-complete even when the input graphs are chordal, planar, line or total graphs. We present an O(m √ n) algorithm for the VDT problem on split graphs and an O(n) algorithm for the VDT problem on cographs. A linear algorithm for the triangle cover problem on strongly chordal graphs is also presented. Finally, the notion of packing-hardness, which may be crucial to the understanding of the difficulty of generalized matching problems, is defined.