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

A note on the independent roman domination in unicyclic graphs

A Roman dominating function (RDF) on a graph $G = (V;E)$ is a function $f : V \to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. The weight of an RDF is the value $f(V(G)) = \sum _{u \in V (G)} f(u)$. An RDF $f$ in a graph $G$ is independent if no two vertices assigned positive values are adjacent. The Roman domination number $\gamma _R (G)$ (respectively, the independent Roman domination number $i_{R}(G)$) is the minimum weight of an RDF (respectively, independent RDF) on $G$. We say that $\gamma _R (G)$ strongly equals $i_R (G)$, denoted by $\gamma _R (G) \equiv i_R (G)$, if every RDF on $G$ of minimum weight is independent. In this note we characterize all unicyclic graphs $G$ with $\gamma _R (G) \equiv i_R (G)$

Trees with Unique Italian Dominating Functions of Minimum Weight

An Italian dominating function, abbreviated IDF, of $G$ is a function $f \colon V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that for every vertex $v \in V(G)$ with $f(v)=0$, we have $\sum_{u \in N(v)} f(u) \ge 2$. That is, either $v$ is adjacent to at least one vertex $u$ with $f(u) = 2$, or to at least two vertices $x$ and $y$ with $f(x) = f(y) = 1$. The Italian domination number, denoted $\gamma_I$(G), is the minimum weight of an IDF in $G$. In this thesis, we use operations that join two trees with a single edge in order to build trees with unique $\gamma_I$-functions

The total co-independent domination number of some graph operations

[EN] A set D of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex of D. The total dominating set D is called a total co-independent dominating set if the subgraph induced by V (G)- D is edgeless. The minimum cardinality among all total co-independent dominating sets of G is the total co-independent domination number of G. In this article we study the total co-independent domination number of the join, strong, lexicographic, direct and rooted products of graphs.I. Peterin was partially supported by ARRS Slovenia under grants P1-0297 and J1-9109; I. G. Yero was partially supported by Junta de Andalucia, FEDER-UPO Research and Development Call, reference number UPO-1263769.Cabrera Martinez, A.; Cabrera García, S.; Peterin, I.; Yero, IG. (2022). The total co-independent domination number of some graph operations. Revista de la Unión Matemática Argentina. 63(1):153-158. https://doi.org/10.33044/revuma.165215315863