Partial Vertex Cover on Graphs of Bounded Degeneracy

In the Partial Vertex Cover (PVC) problem, we are given an $n$-vertex graph $G$ and a positive integer $k$, and the objective is to find a vertex subset $S$ of size $k$ maximizing the number of edges with at least one end-point in $S$. This problem is W[1]-hard on general graphs, but admits a parameterized subexponential time algorithm with running time $2^{O(\sqrt{k})}n^{O(1)}$ on planar and apex-minor free graphs [Fomin et al. (FSTTCS 2009, IPL 2011)], and a $k^{O(k)}n^{O(1)}$ time algorithm on bounded degeneracy graphs [Amini et al. (FSTTCS 2009, JCSS 2011)]. Graphs of bounded degeneracy contain many sparse graph classes like planar graphs, $H$-minor free graphs, and bounded tree-width graphs. In this work, we prove the following results: 1) There is an algorithm for PVC with running time $2^{O(k)}n^{O(1)}$ on graphs of bounded degeneracy which is an improvement on the previous $k^{O(k)}n^{O(1)}$ time algorithm by Amini et al. 2) PVC admits a polynomial compression on graphs of bounded degeneracy, resolving an open problem posed by Amini et al.