Edge-Disjoint packing of stars and cycles

Abstract

We study the parameterized complexity of two graph packing problems, Edge-Disjoint k-Packing of s-Stars and Edge-Disjoint k-Packing of s-Cycles. With respect to the choice of parameters, we show that although the two problems are FPT with both k and s as parameters, they are unlikely to be fixed-parameter tractable when parameterized by only k or only s. In terms of kernelization complexity, we show that Edge-Disjoint k-Packing of s-Stars has a kernel with size polynomial in both k and s, but in contrast, unless NP ⊆ coNP/poly, Edge-Disjoint k-Packing of s-Cycles does not have a kernel with size polynomial in both k and s, and moreover does not have a kernel with size polynomial in s for any fixed k. Specifically, (1) from the negative direction, we show that Edge-Disjoint k-Packing of s-Stars is W[1]-hard with parameter k in general graphs, and that Edge-Disjoint k-Packing of s-Cycles is W[1]-hard with parameter k and NP-hard for any even s≥4 in bipartite graphs; (2) from the positive direction, we show that Edge-Disjoint k-Packing of s-Stars admits a ks2 kernel, and that Edge-Disjoint k-Packing of 4-Cycles admits a 96k2 kernel in general graphs and a 96k kernel in planar graphs.