Abstract
Let G be a graph without isolated vertices. A function f : V ( G ) → { 0 , 1 , 2 } is a total Roman dominating function on G if every vertex v ∈ V ( G ) for which f ( v ) = 0 is adjacent to at least one vertex u ∈ V ( G ) such that f ( u ) = 2 , and if the subgraph induced by the set { v ∈ V ( G ) : f ( v ) ≥ 1 } has no isolated vertices. The total Roman domination number of G, denoted γ t R ( G ) , is the minimum weight ω ( f ) = ∑ v ∈ V ( G ) f ( v ) among all total Roman dominating functions f on G. In this article we obtain new tight lower and upper bounds for γ t R ( G ) which improve the well-known bounds 2 γ ( G ) ≤ γ t R ( G ) ≤ 3 γ ( G ) , where γ ( G ) represents the classical domination number. In addition, we characterize the graphs that achieve equality in the previous lower bound and we give necessary conditions for the graphs which satisfy the equality in the upper bound above.
Highlights
Domination theory is a classical and interesting topic in theory of graphs, as well as one of the most active areas of research in this topic
Abdollahzadeh Ahangar et al [16] give the relationship between the total Roman domination number and the domination number of a graph: If G is a graph with no isolated vertex
New results concerning the study of total Roman domination in graphs have been presented in this article
Summary
Domination theory is a classical and interesting topic in theory of graphs, as well as one of the most active areas of research in this topic. The semitotal domination number, denoted by γt ( G ), is the minimum cardinality among all semitotal dominating sets of G This parameter was introduced by Goddard et al in [5], and was further studied in [6,7,8]. This fact may be because the classical (total) domination problem can be studied using functions defined on graphs Based on this approach, we consider the following concepts, which are variants of the domination in graphs. A total Roman dominating function (TRDF) on a graph G without isolated vertices, is an RDF f (V0 , V1 , V2 ) such that the set V1 ∪ V2 is a total dominating set of G.
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