Eternal Domination Number Research Articles

Eternal domination and clique covering

We study the relationship between the eternal domination number of a graph and its clique cove-ring number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We show that the smallest graph having its eternal domination number less than its clique covering number has 10 vertices. We determine the complete set of 10 -vertex and 11 -vertex graphs having eternal domination numbers less than their clique covering numbers. We show that the smallest triangle-free graph with this property has order 13 , as does the smallest circulant graph. We describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. We also consider planar graphs and cubic graphs. Finally, we show that for any integer k ≥ 2 there exist infinitely many graphs having domination number and eternal domination number equal to k containing dominating sets which are not eternal dominating sets.

The game of Cops and Eternal Robbers

We introduce the game of Cops and Eternal Robbers played on graphs, where there are infinitely many robbers that appear sequentially over distinct plays of the game. A positive integer t is fixed, and the cops are required to capture the robber in at most t time-steps in each play. The associated optimization parameter is the eternal cop number, denoted by ct∞, which equals the eternal domination number in the case t=1, and the cop number for sufficiently large t. We study the complexity of Cops and Eternal Robbers, and show that the game is NP-hard when t is a fixed constant and EXPTIME-complete for large values of t. We determine precise values of ct∞ for paths and cycles. The eternal cop number is studied for retracts, and this approach is applied to give bounds for trees, as well as for strong and Cartesian grids.

Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number \(\gamma ^{\infty }_{all}\) of a graph, which is the minimum number of guards required to defend against an infinite sequence of attacks. This paper first continues the study of the eternal domination game on strong grids \(P_n\boxtimes P_m\). Cartesian grids \(P_n \square P_m\) have been vastly studied with tight bounds existing for small grids such as \(k\times n\) grids for \(k\in \{2,3,4,5\}\). It was recently proven that \(\gamma ^{\infty }_{all}(P_n \square P_m)=\gamma (P_n \square P_m)+O(n+m)\) where \(\gamma (P_n \square P_m)\) is the domination number of \(P_n \square P_m\) which lower bounds the eternal domination number [Lamprou et al. Eternally dominating large grids. Theoretical Computer Science, 794:27–46, 2019]. We prove that, for all \(n,m\in \mathbb {N^*}\) such that \(m\ge n\), \(\lfloor \frac{n}{3} \rfloor \lfloor \frac{m}{3} \rfloor +\Omega (n+m)=\gamma _{all}^{\infty } (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil + O(m\sqrt{n})\) (note that \(\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil\) is the domination number of \(P_n\boxtimes P_m\)). We then generalise our technique to prove that \(\gamma _{all}^{\infty }(G)=\gamma (G)+o(\gamma (G))\) for all graphs \(G\in {\mathcal {F}}\), where \({\mathcal {F}}\) is a large family of D-dimensional grids which are supergraphs of the D-dimensional Cartesian grid and subgraphs of the D-dimensional strong grid. In particular, \({\mathcal {F}}\) includes both the D-dimensional Cartesian grid and the D-dimensional strong grid.

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph G(n,rn), n vertices are placed randomly in Euclidean d-space and edges are added between any pair of vertices distant at most rn from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for G(n,rn) in the thermodynamic limit with nrnd= const., and also in the dense limit with nrnd→∞, rn→0. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the traveling salesman, spanning tree, matching, bipartite matching and bipartite traveling salesman problems, for a general class of weight functions with at most polynomial growth of order d−ε, under thermodynamic scaling of the distance parameter.

Eternal dominating sets on digraphs and orientations of graphs

Eternal domination is a problem that asks the following question: Can we eternally defend a graph ? The principle is to defend against an attacked vertex, that changes every turn, by moving guards along the edges of the graph. In the classical version of the game (Burger et al., 2004), only one guard can move at a time, but in the m-eternal version (Goddard et al., 2005), any number of guards can move in a single turn. This problem led to the introduction of two parameters: the eternal and m-eternal domination numbers of the graph, that are the minimum numbers of guards necessary to defend against any infinite sequence of attacks.In this paper, we first study the (m-)eternal domination number on digraphs. We show that a lot of results proven for undirected graphs can be generalized to digraphs. Then, we introduce the problem of oriented (m-)eternal domination, that consists in finding an orientation of a graph that minimizes the eternal domination number. We prove that computing the oriented eternal domination number is a coNP-hard problem, and we give a complete characterization of the graphs for which the oriented m-eternal domination number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, and different kinds of grids and products of graphs.

Eternal domination on prisms of graphs

An eternal dominating set of a graph G is a set of vertices (or “guards”) which dominates G and which can defend any infinite series of vertex attacks, where an attack is defended by moving one guard along an edge from its current position to the attacked vertex. The size of the smallest eternal dominating set is denoted γ∞(G) and is called the eternal domination number of G. In this paper, we answer a conjecture of Klostermeyer and Mynhardt [Discussiones Mathematicae Graph Theory, vol. 35, pp. 283–300], showing that there exist infinitely many graphs G such that γ∞(G)=θ(G) and γ∞(G□K2)<θ(G□K2), where θ(G) denotes the clique cover number of G.

A method for eternally dominating strong grids

In the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard may occupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number of guards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as $2\times n$, $3\times n$, $4\times n$, and $5\times n$ grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] that the eternal domination number of these grids in general is within $O(m+n)$ of their domination number which lower bounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of strong grids is upper bounded by $\frac{mn}{6}+O(m+n)$. We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the eternal domination number of strong grids is upper bounded by $\frac{mn}{7}+O(m+n)$. While this does not improve upon a recently announced bound of $\lceil\frac{m}{3}\rceil \lceil\frac{n}{3}\rceil+O(m\sqrt{n})$ [Mc Inerney, Nisse, Perennes, CIAC 2019] in the general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is at most $6179$.

Domination and Eternal Domination of Jahangir Graph

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. We consider the “all guards move” of the eternal dominating set problem. In which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current position after each attack. If the new formed set of guards is still a dominating set of the graph then we successfully defended the attack. Our goal is to find the minimum number of guards required to eternally protect the graph. We call this number the m-eternal domination number and we denote it by . In this paper we find the eternal domination number of Jahangir graph Js,m for s=2,3 and arbitrary m. We also find the domination number for J3,m .

A note on the eternal dominating set problem

We consider the “all guards move” model for the eternal dominating set problem. A set of guards form a dominating set on a graph and at the beginning of each round, a vertex not in the dominating set is attacked. To defend against the attack, the guards move (each guard either passes or moves to a neighboring vertex) to form a dominating set that includes the attacked vertex. The minimum number of guards required to defend against any sequence of attacks is the “eternal domination number” of the graph. In 2005, it was conjectured [Goddard et al. (J. Combin. Math. Combin. Comput. 52:169–180, 2005)] there would be no advantage to allow multiple guards to occupy the same vertex during a round. We show this is, in fact, false. We also describe algorithms to determine the eternal domination number for both models for eternal domination and examine the related combinatorial game, which makes use of the reduced canonical form of games.

Bounds for the $m$-Eternal Domination Number of a Graph

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The $m$-eternaldomination number, $\edom(G)$, of a graph $G$ is the minimum number of guards needed to defend $G$ against any such sequence. We show that if $G$ is a connected graph with minimum degree at least~$2$ and of order~$n \ge 5$, then $\edom(G) \le \left\lfloor \frac{n-1}{2} \right\rfloor$, and this bound is tight. We also prove that if $G$ is a cubic bipartite graph of order~$n$, then $\edom(G) \le \frac{7n}{16}$.

Closing the Gap: Eternal Domination on 3 x n Grids

The domination number for grid graphs has been a long studied problem; the first results appeared over thirty years ago [Jacobson 1984] and the final results appeared in 2013 [Goncalves 2013]. Grid graphs are a natural class of graphs to consider for the eternal dominating set problem as the domination number forms a lower bound for the eternal domination number. The 3 x n grid has been considered in several papers, and the difference between the upper and lower bounds for the eternal domination number in the all-guards move model has been reduced to a linear function of n. In this short paper, we provide an upper bound for the eternal domination number which exceeds the lower bound by at most 3.

Eternal domination: criticality and reachability

We show that for every minimum eternal dominating set, D, of a graph G and every vertex v ∈ D, there is a sequence of attacks at the vertices of G which can be defended in such a way that an eternal dominating set not containing v is reached. The study of the stronger assertion that such a set can be reached after a single attack is defended leads to the study of graphs which are critical in the sense that deleting any vertex reduces the eternal domination number. Examples of these graphs and tight bounds on connectivity, edgeconnectivity and diameter are given. It is also shown that there exist graphs in which deletion of any edge increases the eternal domination number, and graphs in which addition of any edge decreases the eternal

Trees with large [formula omitted]-eternal domination number

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The m-eternal domination number, γm∞(G), of a graph G is the minimum number of guards needed to defend G against any such sequence. We characterize the class of trees of order n with maximum possible m-eternal domination number, which is ⌈n2⌉.

Vertex covers and eternal dominating sets

The 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.

Graphs with equal eternal vertex cover and eternal domination numbers

Mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on either its vertices or its edges. If attacks occur at vertices, this is known at the eternal domination problem. If attacks occur at edges, this is known as the eternal vertex cover problem. We focus on the model in which all guards can move to neighboring vertices in response to an attack. Motivated by the question of which graphs have equal eternal vertex cover and eternal domination numbers, a number of results are presented; one of the main results of the paper is that the eternal vertex cover number is greater than the eternal domination number (in the all-guards move model) in all graphs of minimum degree at least two.

Tight bounds for eternal dominating sets in graphs

The eternal domination number of a graph is the number of guards needed at vertices of the graph to defend the graph against any sequence of attacks at vertices. We consider the model in which at most one guard can move per attack and a guard can move across at most one edge to defend an attack. We prove that there are graphs G for which γ ∞ ( G ) ⩾ α ( G ) + 1 2 , where γ ∞ ( G ) is the eternal domination number of G and α ( G ) is the independence number of G. This matches the upper bound proved by Klostermeyer and MacGillivray.