A 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 γ(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])=∑u∈N[v]f(u)≥1 for each v∈V, where N[v] is the closed neighborhood of v. The minus domination number of G is γ−(G)=min{∑v∈Vf(v)∣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 γ−γ− can be made arbitrary large. This note corrects the mistakes in the proof and poses a new problem on the upper bound for γ−γ− in cubic graphs.