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...

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...

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...

A double Roman dominating function of a graph $G$ is $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that for each vertex $v$ with $f(v)=0$, there exists $u\in N(v)$ $f(u)=3$, or are $u,w\in $f(u)=f(w)=2$, and if $f(v)=1$, then adjacent to assigned at least $2$ under $f$. The domination number $\gamma_{dR}(G)$ minimum weight $f(V(G))=\sum_{v\in V(G)}f(v)$ among all functions $G$. An outer i...

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...