Edge-Disjoint Packing of Stars and Cycles

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 \(\subseteq \) 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 \(s \ge 4\) in bipartite graphs; (2) from the positive direction, we show that Edge-Disjoint \(k\)-Packing of \(s\)-Stars admits a \(ks^2\) kernel, and that Edge-Disjoint \(k\)-Packing of 4-Cycles admits a \(96k^2\) kernel in general graphs and a 96k kernel in planar graphs.