Upper minus total domination in small-degree regular graphs

A function f:V(G)→{-1,0,1}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}g:V(G)→{-1,0,1}, f≠gf≠g, for which g(v)⩽f(v)g(v)⩽f(v) for every v∈V(G)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.