Let D=(V(D),A(D)) be a finite, simple digraph and k positive integer. A function f:V(D)→{0,1,2,…,k+1} is called [k]-Roman dominating (for short, [k]-RDF) if f(AN−[v])≥|AN−(v)|+k for any vertex v∈V(D), where AN−(v)={u∈N−(v):f(u)≥1} AN−[v]=AN−(v)∪{v}. The weight of [k]-RDF f ω(f)=∑v∈V(D)f(v). minimum on D the domination number, denoted by γ[kR](D). For k=2 k=3, we call them double Roman number tr...

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...

OF THE DISSERTATION Applications and Variations of Domination in Graphs by Paul Andrew Dreyer, Jr. Dissertation Director: Fred S. Roberts In a graph G = (V, E), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively. This dissertation explores some new variations and applications...

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 ...

Let G be a graph with no isolated vertex and let N(v) the open neighbourhood of v∈V(G). f:V(G)→{0,1,2} function Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is strongly total Roman dominating on if subgraph induced by V1∪V2 has N(v)∩V2≠∅ v∈V(G)\V2. The domination number G, denoted γtRs(G), defined as minimum weight ω(f)=∑x∈V(G)f(x) among all functions G. This paper devoted to study it ...