‎A Roman dominating function (RDF) on a graph G=(V,E) is a function  f : V → {0, 1, 2}  such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over ...

Let G = (V,E) be a graph and f be a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ′ : V → {0, 1, 2} defined by f ′ (u) = 1, f ′ (v) = f...

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. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(...

Let G be a graph with no isolated vertex and f : V ( ) → {0, 1, 2} function. i = { x ∈ } for every . We say that is total Roman dominating function on if in 0 adjacent to at least one 2 the subgraph induced by 1 ∪ has vertex. The weight of ω ∑ v minimum among all functions domination number , denoted γ t R It known general problem computing NP-hard. In this paper, we show H nontrivial graph, th...

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 adjacent to at least one assigned 3 or two vertices 2, and with f(u)=1 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. We obtain closed expressions generalized Petersen graphs P(5k,k). It proven ?dR(P(5k,k))=8k k?2,3mod5 8k??dR(P(5...

For a simple graph G=(V,E) with no isolated vertices, total Roman {3}-dominating function(TR3DF) on G is function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 f(v)=1; and (iii) every vertex v f(v)≠0 has neighbor u f(u)≠0 for v∈V(G). The weight of TR3DF f sum f(V)=∑v∈V(G)f(v) minimum called {3}-domination number denoted by γt{R3}(G). In this paper, we...