A feedback vertex set of 2-degenerate graphs

A feedback vertex set of a graph G is a set S of its vertices such that the subgraph induced by V(G) \ S is a forest. The cardinality of a minimum feedback vertex set of G is denoted by del(G). A graph G is 2-degenerate if each subgraph G' of G has a vertex v such that d(G')(v) <= 2. In this paper, we prove that del(G) <= 2n/5 for any 2-degenerate n-vertex graph G and moreover, we show that this bound is tight. As a consequence, we derive a polynomial time algorithm, which for a given 2-degenerate n-vertex graph returns its feedback vertex set of cardinality at most 2n/5. (C) 2014 Elsevier B.V. All rights reserved.