The K _r -Packing Problem

Abstract

For a fixed integer r≥2, the K _r -packing problem is to find the maximum number of pairwise vertex-disjointK _r 's (complete graphs on r vertices) in a given graph. The K _r -factor problem asks for 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≥3 – it is known that for r≥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≥3 the K _r -packing problem is NP-complete even when restrict to chordal graphs, planar graphs (for r=3, 4 only), line graphs and 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≥4), and for the K _r -packing problem on cographs.