A Roman dominating function (RD-function) on a graph G = ( V , E ) is f : → {0, 1, 2} satisfying the condition that every vertex u for which 0 adjacent to at least one v 2. An in perfect (PRD-function) if with exactly The (perfect) domination number γ R p )) minimum weight of an . We say strongly equals ), denoted by ≡ γR RD-function PRD-function. In this paper we show given it NP-hard decide w...

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(...

A restrained Italian dominating function (RID-function) on a graph G=(V,E) is f:V→{0,1,2} satisfying: (i) f(N(u))≥2 for every vertex u∈V(G) with f(u)=0, where N(u) the set of vertices adjacent to u; (ii) subgraph induced by assigned 0 under f has no isolated vertices. The weight an RID-function sum its value over whole vertices, and domination number minimum G. In this paper, we initiate study ...

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...

Given graph G = (V,E), a dominating set S is a subset of vertex set V such that any vertex not in S is adjacent to at least one vertex in S. The domination number of a graph G is the minimum size of the dominating sets of G. In this paper we study some results on domination number, connected, independent, total and restrained domination number denoted by γ(G), γc(G) ,γi(G), γt(G) and γr(G) resp...

OBJECTIVE To assess the effect of pelvic stabilization and hip position on the electromyographic activity of trunk extensors during Roman chair exercise. A secondary objective was to compare genders. DESIGN Repeated measures. SUBJECTS Eleven men and 11 women volunteers. METHODS Five trunk flexion-extension cycles for 3 Roman chair conditions: (i) pelvis unrestrained; (ii) pelvis restraine...

This paper is devoted to the study of quadruple Roman domination in trees, and it a contribution Special Issue “Theoretical computer science discrete mathematics” Symmetry. For any positive integer k, [k]-Roman dominating function ([k]-RDF) simple graph G from vertex set V {0,1,2,…,k+1} if for u?V with f(u)<k, ?x?N(u)?{u}f(x)?|{x?N(u):f(x)?1}|+k, where N(u) open neighborhood u. The weight [k...

Let G = (V,E) be a graph and let f be a function f : E → {0, 1, 2}. An edge x with f(x) = 0 is said to be undefended with respect to f if it is not incident to an edge with positive weight. The function f is a weak edge Roman dominating function (WERDF) if each edge x with f(x) = 0 is incident to an edge y with f(y) > 0 such that the function f ′ : E → {0, 1, 2}, defined by f ′(x) = 1, f ′(y) =...

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...