Upper Minus Total Domination of a \(5\)-Regular Graph

Abstract

A function \(f: V(G) \to \{-1,0,1\}\) 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. That is, for every \(v \in V\), \(f(N(v)) \geq 1\), where \(N(v)\) consists of every vertex adjacent to \(v\). The weight of a MTDF is the sum of its function values over all vertices. A MTDF \(f\) is minimal if there does not exist a MTDF \(g: V(G) \to \{-1,0,1\}\), \(f \neq g\), for which \(g(v) \leq f(v)\) for every \(v \in V\). The upper minus total domination number, denoted by \(\Gamma^{-}_{t}(G)\), of \(G\) is the maximum weight of a minimal MTDF on \(G\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. The signed total domination number, denoted by \(\gamma^{s}_{t}(G)\), of \(G\) is the minimum weight of a STDF on \(G\). In this paper, we establish an upper bound on \(\Gamma^{-}_{t}(G)\) of the 5-regular graph and characterize the extremal graphs attaining the upper bound. Also, we exhibit an infinite family of cubic graphs in which the difference \(\Gamma^{-}_t(G) – \gamma^{s}_t(G)\) can be made arbitrarily large.