M-Secure vertex cover of a graph

Abstract

In this paper, we use multiple guard movements to defend the edges of a graph [Formula: see text] against a single attack. At most, one guard is positioned at each vertex. To defend an attack on an edge, a guard at an incident vertex moves across the attacked edge and the other guards may move (or not) to the neighboring vertices to better configure themselves. This strategy requires the set of vertices containing guards to be a vertex cover before and after an attack. A suitable placement of guards is called an [Formula: see text]-secure vertex cover of [Formula: see text]. We call this the [Formula: see text]-secure vertex cover problem, where [Formula: see text] stands for the multiple guard movements. The minimum number of guards required to defend the edges of [Formula: see text] against a single attack using multiple guard movements is called the [Formula: see text]-secure vertex covering number and it is denoted by [Formula: see text]. In this paper we initiate a study of this parameter.