A Roman dominating function 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 a Roman dominating function is the value f(V ) = ∑ u∈V f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this pape...

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite 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 twoneighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must haveat least one neighbor $u$ with $f(u)ge 2$. The weight of a double R...

The signed edge domination number of a graph is an edge variant of the signed domination number. The closed neighbourhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end vertex with e. Let f be a mapping of the edge set E(G) of G into the set {−1, 1}. If ∑ x∈N [e] f(x) 1 for each e ∈ E(G), then f is called a signed edge dominating function on G. T...

A Roman dominating function on a graphG is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u ∈ V (G) for which f(u) = 0 is adjacent to at least one vertex v ∈ V (G) for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number γR(G) of G is the minimum weight of a Roman dominating function on G. A Ro...

LetD = (V,A) be a finite and simple digraph. A Roman dominating function (RDF) on a digraph D is a labeling f : V (D) → {0, 1, 2} such that every vertex with label 0 has a in-neighbor with label 2. The weight of an RDF f is the value ω(f) = ∑ v∈V f(v). The Roman domination number of a digraph D, denoted by γR(D), equals the minimum weight of an RDF on D. In this paper we present some sharp boun...

Edge Roman Star Domination Number on Graphs Angshu Kumar Sinha, Akul Rana and Anita Pal Department of Mathematics, NSHM Knowledge Campus Durgapur -713212, INDIA. e-mail: [email protected] Department of Mathematics, Narajole Raj College Narajole, Paschim Medinipur721211, INDIA. e-mail: [email protected] Department of Mathematics, National Institute of Technology Durgapur Durgapur-713209, I...