Roman [1,2]-domination of graphs

Abstract

A [1,2]-set of a graph G is a set S of vertices of G if every vertex not in S has one or two neighbors in S. The [1,2]-domination number γ[1,2](G) of G equals the minimum cardinality of a [1,2]-set of G. A Roman [1,2]-dominating function (R[1,2]DF) on a graph G is a function f from the vertex set V of G to the set {0,1,2} such that any vertex assigned 0 under f has one or two neighbors assigned 2. The weight of an R[1,2]DF f is the sum ∑x∈Vf(x). The Roman [1,2]-domination number γR[1,2](G) of G equals the minimum weight of an R[1,2]DF on G. In this paper, we prove that the decision problem on the Roman [1,2]-domination is NP-complete for bipartite and chordal graphs. Moreover, we give some bounds on the Roman [1,2]-domination number. In particular, we show that for any nontrivial tree T, γR[1,2](T)≥γ[1,2](T)+1 and characterize all trees obtaining equality in this bound.