Let $G=(V,E)$ be a finite and simple graph of order $n$ maximumdegree $\Delta$. A signed strong total Roman dominating function ona $G$ is $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ open neighborhood (ii) every forwhich $f(v)=-1$ adjacent to at least one vertex...

A dominating set D for a graph G is a subset of V (G) such that any vertex not in D has at least one neighbor in D. The domination number γ(G) is the size of a minimum dominating set in G. Vizing’s conjecture from 1968 states that for the Cartesian product of graphs G and H, γ(G)γ(H) ≤ γ(G H), and Clark and Suen (2000) proved that γ(G)γ(H) ≤ 2γ(G H). In this paper, we modify the approach of Cla...

A set D of vertices in a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in D. A total dominating set D of G is said to be weak if every vertex v ∈ V −D is adjacent to a vertex u ∈ D such that dG(v) ≥ dG(u). The weak total domination number γwt(G) of G is the minimum cardinality of a weak total dominating set of G. A total dominating set D of G is said ...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function 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. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...

Pell graphs are defined on certain ternary strings as special subgraphs of Fibonacci cubes odd index. In this work the domination number, total 2-packing connected paired and signed number studied. Using integer linear programming, exact values some estimates for these numbers small obtained. Furthermore, theoretical bounds obtained graphs.

We consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers fi assigned to each node i, such that for any node j in the graph, there exists some node k having a positive fk-value whose shortest distance to node j is no more than fk. The cost of a broadcast...

A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V → {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 f is the value f(V (G)) = ∑ u∈V (G) f(u). A function f : V (G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V (G), where Vi = {v ∈ V (G) | f(v) = i} for i = 0...

A Roman dominating function (RDF) on a graphG = (V,E) is a function f : V −→ {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)) = ∑ u∈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 γR(G)...