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

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

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

The Signed Roman Domatic Number of a Digraph

Let $D$ be a finite and simple digraph with vertex set $V(D)$.A {\em signed Roman dominating function} on the digraph $D$ isa function $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that$\sum_{u\in N^-[v]}f(u)\ge 1$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ andall inner neighbors of $v$, and every vertex $u\in V(D)$ for which $f(u)=-1$ has an innerneighbor $v$ for which $f(v)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signedRoman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each$v\in V(D)$, is called a {\em signed Roman dominating family} (of functions) on $D$. The maximumnumber of functions in a signed Roman dominating family on $D$ is the {\em signed Roman domaticnumber} of $D$, denoted by $d_{sR}(D)$. In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for $d_{sR}(D)$. In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs

The Italian Domatic Number of Varying Graph Families

Submitting PDF to site manager due to formatting issue

Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs