The K-r-packing problem

Abstract

For a fixed integer r greater than or equal to 2, the K-r-packing problem is to find the maximum number of pairwise vertex-disjoint K-r's (complete graphs on r vertices) in a given graph. The K-r-factor problem asks fur the existence of a partition of the vertex set of a graph into K-r's. The K-r-packing problem is a natural generalization of the classical matching problem, but turns out to be much harder for r greater than or equal to 3 - it is known that for r greater than or equal to 3 the K-r-factor problem is NP-complete: for graphs with clique number r [16]. This paper considers the complexity of the K-r-packing problem on restricted classes of graphs. We first prove that for r greater than or equal to 3 the K-r-packing problem is NP-complete even when restrict to chordal graphs, planar graphs (for r = 3. 4 only), line graphs acid total graphs. The hardness result for K-3- packing on chordal graphs answers an open question raised in [6]. We also give (simple) polynomial algorithms for the K-3-packing and the K-r-factor problems on split graphs (this is interesting in light of the fact that K-r-packing becomes NP-complete on split graphs for r greater than or equal to 4), and for the K-r-packing problem on cographs.