A note on domination and minus domination numbers in cubic graphs Original Research Article

Let G=(V,E)G=(V,E) be a graph. A subset SS of VV is called a dominating set if each vertex of V−SV−S has at least one neighbor in SS. The domination number γ(G)γ(G) equals the minimum cardinality of a dominating set in GG. A minus dominating function on GG is a function f:V→{−1,0,1}f:V→{−1,0,1} such that f(N[v])=∑u∈N[v]f(u)≥1f(N[v])=∑u∈N[v]f(u)≥1 for each v∈Vv∈V, where N[v]N[v] is the closed neighborhood of vv. The minus domination number of GG is γ−(G)=min{∑v∈Vf(v)∣f is a minus dominating function on G}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−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−γ−γ− in cubic graphs.