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

Trees whose 2-domination subdivision number is 2

A set $S$ of vertices in a graph $G = (V,E)$ is a $2$-dominating set if every vertex of $V\setminus S$ is adjacent to at least two vertices of $S$. The $2$-domination number of a graph $G$, denoted by $\gamma_2(G)$, is the minimum size of a $2$-dominating set of $G$. The $2$-domination subdivision number $sd_{\gamma_2}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the $2$-domination number. The authors have recently proved that for any tree $T$ of order at least $3$, $1 \leq sd_{\gamma_2}(T)\leq 2$. In this paper we provide a constructive characterization of the trees whose $2$-domination subdivision number is $2$

Domination parameters with number 2: interrelations and algorithmic consequences

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak $2$-domination number, $\gamma_{w2}(G)$, the $2$-domination number, $\gamma_2(G)$, the $\{2\}$-domination number, $\gamma_{\{2\}}(G)$, the double domination number, $\gamma_{\times 2}(G)$, the total $\{2\}$-domination number, $\gamma_{t\{2\}}(G)$, and the total double domination number, $\gamma_{t\times 2}(G)$, where $G$ is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, $\gamma_R(G)$, and two classical parameters, the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$, we consider 13 domination invariants in graphs $G$. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain some complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure