Partial Vertex Cover on Graphs of Bounded Degeneracy

Panolan, Fahad and Yaghoubizade, Hannane (2022) Partial Vertex Cover on Graphs of Bounded Degeneracy. In: 17th International Computer Science Symposium in Russia, CSR 2022, 29 June 2022through 1 July 2022, Virtual, Online.

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Abstract

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 2O(k)nO(1) on planar and apex-minor free graphs [Fomin et al. (FSTTCS 2009, IPL 2011)], and a kO(k)nO(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 (see Fig. 1). In this work, we prove the following results: There are algorithms for PVC on graphs of degeneracy d with running time 2 O(dk)nO(1) and (e+ ed) k2 o(k)nO(1) which are improvements on the previous kO(k)nO(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. [2]. © 2022, Springer Nature Switzerland AG.

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