Vertex Cover at Distance on H-Free Graphs

Abstract

AbstractThe question of characterizing graphs H such that the Vertex Cover problem is solvable in polynomial time in the class of H-free graphs is notoriously difficult and still widely open. We completely solve the corresponding question for a distance-based generalization of vertex cover called distance-k vertex cover, for any positive integer k. In this problem the task is to determine, given a graph G and an integer \(\ell \), whether G contains a set of at most \(\ell \) vertices such that each edge of G is at distance at most k from a vertex in the set. We show that for all \(k \ge 1\) and all graphs H, the distance-k vertex cover problem is solvable in polynomial time in the class of H-free graphs if H is an induced subgraph of \(P_{2k+2} + sP_{\max \{k,2\}}\) for some \(s \ge 0\), and NP-complete otherwise. KeywordsDistance-k Vertex CoverH-free graphNP-completenessPolynomial-time algorithmDichotomy