Graphs without two vertex-disjoint S-cycles

Abstract

Lovasz (1965) characterized graphs without two vertex-disjoint cycles, which implies that such graphs have at most three vertices hitting all cycles. In this paper, we ask whether such a small hitting set exists for S-cycles, when a graph has no two vertex-disjoint S-cycles. For a graph G and a vertex set S of G, an S-cycle is a cycle containing a vertex of S.,We provide an example G on 21 vertices where G has no two vertex-disjoint S-cycles, but three vertices are not sufficient to hit all S-cycles. On the other hand, we show that four vertices are enough to hit all S-cycles whenever a graph has no two vertex-disjoint S-cycles. (C) 2020 Elsevier B.V. All rights reserved.,