A parameterized perspective on packing paths of length two

Abstract

We study (vertex-disjoint) packings of paths of length two (i.e., of P 2’s) in graphs under a parameterized perspective. Starting from a maximal P 2-packing ℘ of size j we use extremal combinatorial arguments for determining how many vertices of ℘ appear in some P 2-packing of size (j+1) (if such a packing exists). We prove that one can ‘reuse’ 2.5j vertices. We also show that this bound is asymptotically sharp. Based on a WIN-WIN approach, we build an algorithm which decides, given a graph, if a P 2-packing of size at least k exists in time $\mathcal{O}^{*}(2.448^{3k})$ .