On [k] -Roman domination in graphs

Abstract

For an integer k ≥ 1 , let f be a function that assigns labels from the set { 0 , 1 , … , k + 1 } to the vertices of a simple graph G = ( V , E ) . The active neighborhood AN(v) of a vertex v ∈ V ( G ) with respect to f is the set of all neighbors of v that are assigned non-zero values under f. A [ k ] -Roman dominating function ( [ k ] -RDF) is a function f : V ( G ) → { 0 , 1 , 2 , … , k + 1 } such that for every vertex v ∈ V ( G ) with f(v) < k, we have ∑ u ∈ N [ v ] f ( u ) ≥ | A N ( v ) | + k . The weight of a [ k ] -RDF is the sum of its function values over the whole set of vertices, and the [ k ] -Roman domination number γ [ k R ] ( G ) is the minimum weight of a [ k ] -RDF on G. In this paper we determine various bounds on the [ k ] -Roman domination number. In particular, we show that for any integer k ≥ 2 every connected graph G of order n ≥ 3 , satisfies γ [ k R ] ( G ) ≤ ( 2 k + 1 ) 4 n , and we characterize the graphs G attaining this bound. Moreover, we show that if T is a nontrivial tree, then γ [ k R ] ( T ) ≥ k γ ( T ) + 1 for every integer k ≥ 2 and we characterize the trees attaining the lower bound. Finally, we prove the NP-completeness of the [ k ] -Roman domination problem in bipartite and chordal graphs.