Restrained reinforcement number in graphs

A set $S$ of vertices is a restrained dominating set of a graph $G=(V,E)$ if every vertex in $V\setminus S$ has a neighbor in $S$ and a neighbor in $V\setminus S$. The minimum cardinality of a restrained dominating set is the restrained domination number $\gamma_{r}(G)$. In this paper we initiate the study of the restrained reinforcement number $r_{r}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges $F\subseteq E(\overline{G})$ for which $\gamma _{r}(G+F

Integer Programming Formulations and Probabilistic Bounds for Some Domination Parameters

In this paper, we further study the concepts of hop domination and 2-step domination and introduce the concepts of restrained hop domination, total restrained hop domination, 2-step restrained domination, and total 2-step restrained domination in graphs. We then construct integer programming formulations and present probabilistic upper bounds for these domination parameters

Roman Domination in Complementary Prisms

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect match- ing between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V(G) → {0,1,2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V ) = Σv∈V f(v) over all such functions of G. We study the Roman domination number of complementary prisms. Our main results show that γR(GG) takes on a limited number of values in terms of the domination number of GG and the Roman domination numbers of G and G

RESTRAINED ROMAN REINFORCEMENT NUMBER IN GRAPHS

A restrained Roman dominating function (RRD-function) on a graph $G=(V,E)$ is a function $f$ from $V$ into $\{0,1,2\}$ satisfying: (i)  every vertex $u$ with $f(u)=0$ is adjacent to a vertex $v$ with $f(v)=2$; (ii) the subgraph induced by the vertices assigned 0 under $f$ has no isolated vertices. The weight of an RRD-function is the sum of its function value over the whole set of vertices, and the restrained Roman domination number is the minimum weight of an RRD-function on $G.$ In this paper, we begin the study of the restrained Roman reinforcement number $r_{rR}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges that we must add to the graph to decrease its restrained Roman domination number. We first show that the decision problem associated with the restrained Roman reinforcement problem is NP-hard. Then several properties as well as some sharp bounds of the restrained Roman reinforcement number are presented. In particular it is established that $r_{rR}(T)=1$ for every tree $T$ of order at least three

Italian Domination in Complementary Prisms

Let $G$ be any graph and let $\overline{G}$ be its complement. The complementary prism of $G$ is formed from the disjoint union of a graph $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. An Italian dominating function on a graph $G$ is a function such that $f \, : \, V \to \{ 0,1,2 \}$ and for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N(v)} f(u) \geq 2$. The weight of an Italian dominating function is the value $f(V)=\sum_{u \in V(G)}f(u)$. The minimum weight of all such functions on $G$ is called the Italian domination number. In this thesis we will study Italian domination in complementary prisms. First we will present an error found in one of the references. Then we will define the small values of the Italian domination in complementary prisms, find the value of the Italian domination number in specific families of graphs complementary prisms, and conclude with future problems

On the {2}-domination number of graphs

[EN] Let G be a nontrivial graph and k ¿ 1 an integer. Given a vector of nonnegative integers w = (w0,...,wk), a function f : V(G) ¿ {0,..., k} is a w-dominating function on G if f(N(v)) ¿ wi for every v ¿ V(G) such that f(v) = i. The w-domination number of G, denoted by ¿w(G), is the minimum weight ¿(f) = ¿v¿V(G) f(v) among all w-dominating functions on G. In particular, the {2}- domination number of a graph G is defined as ¿{2} (G) = ¿(2,1,0) (G). In this paper we continue with the study of the {2}-domination number of graphs. In particular, we obtain new tight bounds on this parameter and provide closed formulas for some specific families of graphs.Cabrera-Martínez, A.; Conchado Peiró, A. (2022). On the {2}-domination number of graphs. AIMS Mathematics. 7(6):10731-10743. https://doi.org/10.3934/math.202259910731107437

Bharath hub number of graphs

The mathematical model of a real world problem is designed as Bharath hub number of graphs. In this paper, we study the graph theoretic properties of this variant. Also, we give results for Bharath hub number of join and corona of two connected graphs, cartesian product and lexicographic product of some standard graphs.Publisher's Versio