﻿ Bounds on the restrained Roman domination number of a graph

# Bounds on the restrained Roman domination number of a graph

##### چکیده

A {em Roman dominating function} on a graph \$G\$ is a function\$f:V(G)rightarrow {0,1,2}\$ satisfying the condition that everyvertex \$u\$ for which \$f(u) = 0\$ is adjacent to at least one vertex\$v\$ for which \$f(v) =2\$. {color{blue}A {em restrained Roman dominating}function} \$f\$ is a {color{blue} Roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} The weight of a restrained Roman dominating function isthe value \$omega(f)=sum_{uin V(G)} f(u)\$. The minimum weight of arestrained Roman dominating function of \$G\$ is called the { emrestrained Roman domination number} of \$G\$ and denoted by \$gamma_{rR}(G)\$.In this paper we establish some sharp bounds for this parameter.

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## bounds on the restrained roman domination number of a graph

a {em roman dominating function} on a graph \$g\$ is a function\$f:v(g)rightarrow {0,1,2}\$ satisfying the condition that everyvertex \$u\$ for which \$f(u) = 0\$ is adjacent to at least one vertex\$v\$ for which \$f(v) =2\$. {color{blue}a {em restrained roman dominating}function} \$f\$ is a {color{blue} roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} the wei...

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## On the Roman Edge Domination Number of a Graph

For an integer n ≥ 2, let I ⊂ {0, 1, 2, · · · , n}. A Smarandachely Roman sdominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a function f : V → {0, 1, 2, · · · , n} satisfying the condition that |f(u)− f(v)| ≥ s for each edge uv ∈ E with f(u) or f(v) ∈ I . Similarly, a Smarandachely Roman edge s-dominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a func...

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## On the restrained Roman domination in graphs

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

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## roman game domination subdivision number of a graph

a {em roman dominating function} on a graph \$g = (v ,e)\$ is a function \$f : vlongrightarrow {0, 1, 2}\$ satisfying the condition that every vertex \$v\$ for which \$f (v) = 0\$ is adjacent to at least one vertex \$u\$ for which \$f (u) = 2\$. the {em weight} of a roman dominating function is the value \$w(f)=sum_{vin v}f(v)\$. the roman domination number of a graph \$g\$, denoted by \$gamma_r(g)\$, equals the...

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## Upper bounds for the Roman domination subdivision number of a graph

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

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عنوان ژورنال:

دوره 1  شماره 1

صفحات  75- 82

تاریخ انتشار 2016-06-01

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