The Minimum Rooted-Cycle Cover Problem

Abstract

Given an undirected rooted graph, a cycle containing the root vertex is called a rooted cycle. We study the combinatorial duality between vertex-covers of rooted-cycles, which generalize classical vertex-covers, and packing of disjoint rooted cycles, where two rooted cycles are vertex-disjoint if their only common vertex is the root node. We use Menger’s theorem to provide a characterization of all rooted graphs such that the maximum number of vertex-disjoint rooted cycles equals the minimum size of a subset of non-root vertices intersecting all rooted cycles, for all subgraphs.