Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from {0, 1, 2}. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give O(C) time algorithm for computing Roman domination number of special classes of polygrap...

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 ...

A Roman dominating function on a graph G = (V,E) is a function f : V −→ {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 is the value w(f) = ∑ v∈V f(v). The Roman domination number of a graph G, denoted by γR(G), equals the minimum weight of a Roman dominating function on ...

Given a graph $G=(V,E)$ and a vertex $v in V$, by $N(v)$ we represent the open neighbourhood of $v$. Let $f:Vrightarrow {0,1,2}$ be a function on $G$. The weight of $f$ is $omega(f)=sum_{vin V}f(v)$ and let $V_i={vin V colon f(v)=i}$, for $i=0,1,2$. The function $f$ is said to bebegin{itemize}item a Roman ${2}$-dominating function, if for every vertex $vin V_0$, $sum_{uin N(v)}f(u)geq 2$. The R...

A Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex x with f (x) = 0 is adjacent to at least one vertex y with f (y) = 2. The weight of a Roman dominating function is defined to be f (V ) = ∑ x∈V f (x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we first answer ...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = ∑ v∈V f(v). The k-distance Roman domination number ...

A subset X of edges of a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X . The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. An edge Roman dominating function of a graph G is a function f : E(G) → {0, 1, 2} such that every edge e with f(e) = 0 is adjacent to some edge e′ with f(e′) = 2. T...

In his article published in 1999, Ian Stewart discussed a strategy of Emperor Constantine for defending the Roman Empire. Motivated by this article, Cockayne et al.(2004) introduced the notion of Roman domination in graphs. Let G = (V,E) be a graph. A Roman dominating function of G is a function f : V → {0, 1, 2} such that every vertex v for which f(v) = 0 has a neighbor u with f(u) = 2. The we...

A function f : V (G) → {0, 1, 2} is a Roman dominating function if for every vertex with f(v) = 0, there exists a vertex w ∈ N(v) with f(w) = 2. We introduce two fractional Roman domination parameters, γR ◦ f and γRf , from relaxations of two equivalent integer programming formulations of Roman domination (the former using open neighborhoods and the later using closed neighborhoods in the Roman...

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 matching 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) ofG is the mini...