Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Eternal Independent Sets in Graphs

The use of mobile guards to protect a graph has received much attention in the literature of late in the form of eternal dominating sets, eternal vertex covers and other models of graph protection. In this paper, eternal independent sets are introduced. These are independent sets such that the following can be iterated forever: a vertex in the independent set can be replaced with a neighboring vertex and the resulting set is independent

Perpetually Dominating Large Grids

In the m-\emph{Eternal Domination} game, a team of guard tokens initially occupies a dominating set on a graph $G$. An attacker then picks a vertex without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked vertex, while each remaining one can choose to move to one of his neighboring vertices. The new guards' placement must again be dominating. This attack-defend procedure continues eternally. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise, the attacker wins. The m-\emph{eternal domination number} for a graph $G$ is the minimum amount of guards such that they win against any attacker strategy in $G$ (all guards move model). We study rectangular grids and provide the first known general upper bound on the m-eternal domination number for these graphs. Our novel strategy implements a square rotation principle and eternally dominates $m \times n$ grids by using approximately $\frac{mn}{5}$ guards, which is asymptotically optimal even for ordinary domination.Comment: latest full draft versio

Vertex covers and eternal dominating sets

AbstractThe eternal domination problem requires a graph to be protected against an infinitely long sequence of attacks on vertices by guards located at vertices, the configuration of guards inducing a dominating set at all times. An attack at a vertex with no guard is defended by sending a guard from a neighboring vertex to the attacked vertex. We allow any number of guards to move to neighboring vertices at the same time in response to an attack. We compare the eternal domination number with the vertex cover number of a graph. One of our main results is that the eternal domination number is less than the vertex cover number of any graph of minimum degree at least two having girth at least nine

Spartan Bipartite Graphs Are Essentially Elementary

We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph G and is denoted by evc(G). It is clear that evc(G) is at least mvc(G), the size of a minimum vertex cover of G. We say that G is Spartan if evc(G) = mvc(G). The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on 2n vertices where every edge belongs to a perfect matching, an easy strategy is to have n guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs

Eternal Vertex Cover on Bipartite and Co-Bipartite Graphs

Eternal Vertex Cover problem is a dynamic variant of the vertex cover problem. We have a two player game in which guards are placed on some vertices of a graph. In every move, one player (the attacker) attacks an edge. In response to the attack, the second player (defender) moves the guards along the edges of the graph in such a manner that at least one guard moves along the attacked edge. If such a movement is not possible, then the attacker wins. If the defender can defend the graph against an infinite sequence of attacks, then the defender wins. The minimum number of guards with which the defender has a winning strategy is called the Eternal Vertex Cover Number of the graph G. On general graphs, the computational problem of determining the minimum eternal vertex cover number is NP-hard and admits a 2-approximation algorithm and an exponential kernel. The complexity of the problem on bipartite graphs is open, as is the question of whether the problem admits a polynomial kernel. We settle both these questions by showing that Eternal Vertex Cover is NP-hard and does not admit a polynomial compression even on bipartite graphs of diameter six. We also show that the problem admits a polynomial time algorithm on the class of cobipartite graphs.Comment: 38 pages, 15 figures. Updated to remove a previously incorrect claim about the complexity of the problem on split graph

Perpetually Dominating Large Grids

In the Eternal Domination game, a team of guard tokens initially occupies a dominating set on a graph G. A rioter then picks a node without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked node, while each remaining one can choose to move to one of his neighboring nodes. The new guards' placement must again be dominating. This attack-defend procedure continues perpetually. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise the rioter wins. We study rectangular grids and provide the first known general upper bound for these graphs. Our novel strategy implements a square rotation principle and eternally dominates m x n grids by using approximately (mn)/5 guards, which is asymptotically optimal even for ordinary domination