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 ) = P 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 answer an open pr...

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 every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, . . . , fd} of k-distance Roman dominating functions on G with the property that ∑d i=1 fi(v) ≤ 2 for each v ∈ V (G), is call...

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 Roman dominating function on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value f(G) = ∑ u∈V f(u). The Roman domination number of G is the minimum weight of a Roman dominating function on G. The Roman bondage number of a nonempty ...

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...

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...

Mystery cults were the religious cults within the Greco-Roman world that were secretive in nature but offered their followers the notion of life after death. Mystery cults that threatened Roman political life and the state religion faced suppression and persecution. The mystery cults of Bacchus and Isis were the cults that the Roman government attacked the most. The persecution happened for ent...

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 ...

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

A Roman dominating function on a graph G is a function f : V (G) → {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 (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. In this paper, we s...