A CORRECTION TO A PAPER ON ROMAN κ-DOMINATION IN GRAPHS

Let G = (V,E) be a graph and k be a positive integer. A k-dominating set of G is a subset S ⊆ V such that each vertex in V \S has atleast k neighbors in S. A Roman k-dominating function on G is a functionf : V → {0,1,2} such that every vertex v with f(v) = 0 is adjacent toat least k vertices v 1 ,v 2 ,...,v k with f(v i ) = 2 for i = 1,2,...,k. In thepaper titled “Roman k-domination in graphs” (J. Korean Math. Soc. 46(2009), no. 6, 1309–1318) K. Kammerling and L. Volkmann showed thatfor any graph G with n vertices, γ kR (G) + γ kR (G) ≥ min {2n,4k + 1},and the equality holds if and only if n ≤ 2k or k ≥ 2 and n = 2k + 1or k = 1 and G or G has a vertex of degree n − 1 and its complementhas a vertex of degree n − 2. In this paper we find a counterexampleof Kammerling and Volkmann’s result and then give a correction to theresult. 1. IntroductionLet G = (V,E) be agraphwith vertexset V = V (G) andedge set E = E(G).A k-dominatingsetof G is a subset S ⊆ V such that every vertex in V \S has atleast k neighbors in S. The k-domination numberγ