Roman [1,2]-domination of graphs

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 ∈ V f ( x ). The Roman [ 1 , 2 ]-domination number γ R [ 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 , γ R [ 1 , 2 ] ( T ) ≥ γ [ 1 , 2 ] ( T ) + 1 and characterize all trees obtaining equality in this bound. • The decision problem on Roman [1,2]-domination is NP-complete for bipartite and chordal graphs. • Some bounds on the Roman [1,2]-domination number are given. • The Roman [1,2]-domination number of any nontrivial tree is no less than its [1,2]-domination number plus one.