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 $\gamma _R (G) \equiv i_R (G)$

On a Vizing-type integer domination conjecture

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $\gamma_{\{k\}}(G)$, is defined to be the minimum weight taken over all $\{k\}$-domination functions. Bre\v{s}ar, Henning, and Klav\v{z}ar (On integer domination in graphs and Vizing-like problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $k\ge 2$ so that $\gamma_{\{k\}}(G\square H)\ge \gamma(G)\gamma(H)$. In this note we use the Roman $\{2\}$-domination number, $\gamma_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman $\{2\}$-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $\gamma_{\{2\}}(G\square H)\ge \gamma_{R2}(G\square H)\ge \gamma(G)\gamma(H)$, which also implies the conjecture for all $k\ge 2$.Comment: 8 page

Double Roman domination and domatic numbers of graphs

A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in \cite{bhh} as a function‎ ‎$f:V(G)\rightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$‎, ‎then the vertex $v$ must have at least two‎ ‎neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$‎, ‎and if $f(v)=1$‎, ‎then the vertex $v$ must have‎ ‎at least one neighbor $u$ with $f(u)\ge 2$‎. ‎The weight of a double Roman dominating function $f$ is the sum‎ ‎$\sum_{v\in V(G)}f(v)$‎, ‎and the minimum weight of a double Roman dominating function on $G$ is the double Roman‎ ‎domination number $\gamma_{dR}(G)$ of $G$‎. ‎A set $\{f_1,f_2,\ldots,f_d\}$ of distinct double Roman dominating functions on $G$ with the property that‎ ‎$\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(G)$ is called in \cite{v} a double Roman dominating family (of functions)‎ ‎on $G$‎. ‎The maximum number of functions in a double Roman dominating family on $G$ is the double Roman domatic number‎ ‎of $G$‎. ‎In this note we continue the study of the double Roman domination and domatic numbers‎. ‎In particular‎, ‎we present‎ ‎a sharp lower bound on $\gamma_{dR}(G)$‎, ‎and we determine the double Roman domination and domatic numbers of some‎ ‎classes of graphs

A note on k-Roman graphs

Let $G=\left(V,E\right)$ be a graph and let $k$ be a positive integer. A subset $D$ of $V\left( G\right)$ is a $k$-dominating set of $G$ if every vertex in $V\left( G\right) \backslash D$ has at least $k$ neighbours in $D$. The $k$-domination number $\gamma_{k}(G)$ is the minimum cardinality of a $k$-dominating set of $G.$ A Roman $k$-dominating function on $G$ is a function $f\colon V(G)\longrightarrow\{0,1,2\}$ such that every vertex $u$ for which $f(u)=0$ is adjacent to at least $k$ vertices $v_{1},v_{2},\ldots ,v_{k}$ with $f(v_{i})=2$ for $i=1,2,\ldots ,k.$ The weight of a Roman $k$-dominating function is the value $f(V(G))=\sum_{u\in V(G)}f(u)$ and the minimum weight of a Roman $k$-dominating function on $G$ is called the Roman $k$-domination number $\gamma_{kR}\left( G\right)$ of $G$. A graph $G$ is said to be a $k$-Roman graph if $\gamma_{kR}(G)=2\gamma_{k}(G).$ In this note we study $k$-Roman graphs

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

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

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, γw2(G), the 2-domination number, γ2(G), the {2}-domination number, γ{2}(G), the double domination number, γ×2(G), the total {2}-domination number, γt{2}(G), and the total double domination number, γt×2(G), where G is a graph in which the 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, γR(G), and two classical parameters, the domination number, γ(G), and the total domination number, γt(G), we consider 13 domination invariants in graphs. 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, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Brešar, Boštjan. Institute of Mathematics, Physics and Mechanics; Eslovenia. University of Maribor; EsloveniaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Milanič, Martin. University of Primorska; EsloveniaFil: Safe, Martin Dario. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentin