Note on domination and minus domination numbers in cubic graphs

Let G = (V, E) be a graph. A subset S of V is called a dominating set if each vertex of V - S has at least one neighbor in S. The domination number gamma (G) equals the minimum cardinality of a dominating set in G. A minus dominating function on G is a function f : V -> {-1, 0, 1} such that f(N[v]) = Sigma(u is an element of N[v]) f(u) >= 1 for each v is an element of V, where N[v] is the closed neighborhood of v. The minus domination number of G is gamma(-)(G) = min{Sigma(v is an element of V) f(v) vertical bar f is a minus dominating function on G). It was incorrectly shown in [X. Yang, Q. Hou, X. Huang, H. Xuan, The difference between the domination number and minus domination number of a cubic graph, Applied Mathematics Letters 16 (2003) 1089-1093] that there is an infinite family of cubic graphs in which the difference gamma - gamma(-) can be made arbitrary large. This note corrects the mistakes in the proof and poses a new problem on the upper bound for gamma - gamma(-) in cubic graphs. (c) 2005 Elsevier Ltd. All rights reserved.