On the [1, 2]-domination number of generalized Petersen graphs

Abstract A dominating set in a graph G = ( V , E ) is a subset S of V such that N [ S ] = V , that is, each vertex of G either belongs to S or is adjacent to at least one vertex in S. The minimum cardinality of a dominating set in G is called the domination number, denoted by γ(G). A subset S of V is a [1,2]-set if, for every vertex v ∈ V∖S, v is adjacent to at least one but no more than two vertices in S. The [1,2]-domination number of a graph G, denoted by γ[1, 2](G), is the minimum cardinality of a [1, 2]-set of Chellali et al. gave some bounds for γ[1, 2](G) and proposed the following problem: which graphs satisfy γ ( G ) = γ [ 1 , 2 ] ( G ) . Ebrahimi et al. determined the exact value of the domination number for generalized Petersen graphs P(n, k) when k ∈ {1, 2, 3}. In this paper, we determine the exact values of γ[1, 2](P(n, k)) for k ∈ {1, 2, 3}. We also show that γ [ 1 , 2 ] ( P ( n , k ) ) = γ ( P ( n , k ) ) for k = 1 and k = 3 , respectively, while for k = 2 , γ[1, 2](P(n, k)) ≠ γ(P(n, k)) except for n = 6 , 7 , 9 , 12 .