Abstract
A dominating set in a graph G is a set of vertices S ⊆ V ( G ) such that any vertex of V − S is adjacent to at least one vertex of S . A dominating set S of G is said to be a perfect dominating set if each vertex in V − S is adjacent to exactly one vertex in S. The minimum cardinality of a perfect dominating set is the perfect domination number γ p ( G ) . A function f : V ( G ) → { 0 , 1 , 2 } is a perfect Roman dominating function (PRDF) on G if every vertex u ∈ V for which f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γ R p ( G ) . In this paper, we prove that for any nontrivial tree T, γ R p ( T ) ≥ γ p ( T ) + 1 and we characterize all trees attaining this bound.
Highlights
In this paper, only simple and undirected graph without isolated vertices will be considered.The set of vertices of the graph G is denoted by V = V ( G ) and the edge set is E = E( G )
The order of a graph G is the number of vertices of the graph G and it is denoted by n = n( G )
A dominating set (DS) in a graph G is a set of vertices S ⊆ V ( G ) such that any vertex of V − S is adjacent to at least one vertex of S
Summary
Only simple and undirected graph without isolated vertices will be considered. The set of vertices of the graph G is denoted by V = V ( G ) and the edge set is E = E( G ). A dominating set (DS) in a graph G is a set of vertices S ⊆ V ( G ) such that any vertex of V − S is adjacent to at least one vertex of S. A perfect Roman dominating function (PRDF) on a graph G is an RDF f such that every vertex assigned a 0 is adjacent to exactly one vertex assigned a 2 under f. It is worth mentioning that if S is a minimum (perfect) dominating set of a graph G, clearly (V − S, ∅, S) is a (perfect) RDF and γR ( G ) ≤ 2γ( G ). Motivated by the above example, we shall show in this paper that γR ( T ) ≥ γ p ( T ) + 1 for every nontrivial tree T, and we characterize all trees attaining this bound
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