Stability of Vertex Covers in a Game with Finitely Many Steps

The eternal vertex cover problem is a version of the graph vertex cover problem that canbe represented as a dynamic game between two players (the Attacker and the Defender) withan infinite number of steps. At each step, there is an arrangement of guards over the vertices ofthe graph forming a vertex cover. When the Attacker attacks one of the graph’s edges, theDefender must move the guard along the attacked edge from one vertex to the other. In addition,the Defender can move any number of other guards from their current vertices to some adjacentones to obtain a new vertex cover. The Attacker wins if the Defender cannot build a new vertexcover after the attack.

In this paper, we propose a procedure that allows us to answer the questionwhether there exists a winning Defender strategy that permits protecting a given vertex cover fora given finite number of steps. To construct the Defender strategy, the problem is represented asa dynamic Stackelberg game in which at each step the interaction of the opposing sides isformalized as a two-level mathematical programming problem. The idea of the procedure is torecursively check the 1-stability of vertex covers obtained as a result of solving lower-levelproblems and to use some information about the covers already considered.