Abstract
A function f : V ( G ) → { - 1 , 0 , 1 } defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g : V ( G ) → { - 1 , 0 , 1 } , f ≠ g , for which g ( v ) ⩽ f ( v ) for every v ∈ V ( G ) . The weight of an MTDF is the sum of its function values over all vertices. The minus total domination number of G is the minimum weight of an MTDF on G, while the upper minus domination number of G is the maximum weight of a minimal MTDF on G. In this paper we present upper bounds on the upper minus total domination number of a cubic graph and a 4-regular graph and characterize the regular graphs attaining these upper bounds.
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