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

Disjoint Dominating Sets with a Perfect Matching

In this paper, we consider dominating sets $D$ and $D'$ such that $D$ and $D'$ are disjoint and there exists a perfect matching between them. Let $DD_{\textrm{m}}(G)$ denote the cardinality of smallest such sets $D, D'$ in $G$ (provided they exist, otherwise $DD_{\textrm{m}}(G) = \infty$). This concept was introduced in [Klostermeyer et al., Theory and Application of Graphs, 2017] in the context of studying a certain graph protection problem. We characterize the trees $T$ for which $DD_{\textrm{m}}(T)$ equals a certain graph protection parameter and for which $DD_{\textrm{m}}(T) = \alpha(T)$, where $\alpha(G)$ is the independence number of $G$. We also further study this parameter in graph products, e.g., by giving bounds for grid graphs, and in graphs of small independence number

Bondage number of grid graphs

The bondage number $b(G)$ of a nonempty graph $G$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. Here we study the bondage number of some grid-like graphs. In this sense, we obtain some bounds or exact values of the bondage number of some strong product and direct product of two paths.Comment: 13 pages. Discrete Applied Mathematics, 201

Variable Neighborhood Search Approach for Solving Roman and Weak Roman Domination Problems on Graphs

In this paper Roman and weak Roman domination problems on graphs are considered. Given that both problems are NP hard, a new heuristic approach, based on a Variable Neighborhood Search (VNS), is presented. The presented algorithm is tested on instances known from the literature, with up to 600 vertices. The VNS approach is justified since it was able to achieve an optimal solution value on the majority of instances where the optimal solution value is known. Also, for the majority of instances where optimization solvers found a solution value but were unable to prove it to be optimal, the VNS algorithm achieves an even better solution value

Signed double Roman domination on cubic graphs

The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one $3$-labeled neighbor or at least two $2$-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order $n$ for which we present a sharp $n/2+\Theta(1)$ lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic $2\times m$ grid graphs, among other results