Eternal Connected Vertex Cover Problem

Abstract

A new variant of the classic problem of Vertex Cover (VC) is introduced. Either Connected Vertex Cover (CVC) or Eternal Vertex Cover (EVC) is already a variant of VC, and Eternal Connected Vertex Cover (ECVC) is an eternal version of CVC, or a connected version of EVC. A connected vertex cover of a connected graph G is a vertex cover of G inducing a connected subgraph in G. CVC is the problem of computing a minimum connected vertex cover. For EVC a multi-round game on G is considered, and in response to every attack on some edge e of G, a guard positioned on a vertex of G must cross e to repel it. EVC asks the minimum number of guards to be placed on the vertices of a given G, that is sufficient to repel any sequence of edge attacks of an arbitrary length.