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 $v\in V$, $f(N(v))\ge 1$, 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 $g:V(G)\to \{-1,0,1\}$, $f\ne g$, for which $g(v)\le f(v)$ for every $v\in V$. The upper minus total domination number, denoted by ${\mathrm{\Gamma }}_t^{-}(G)$, 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 ${\gamma }_t^s(G)$, of $G$ is the minimum weight of a STDF on $G$. In this paper, we establish an upper bound on ${\mathrm{\Gamma }}_t^{-}(G)$ 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 ${\mathrm{\Gamma }}_t^{-}(G)–{\gamma }_t^s(G)$ can be made arbitrarily large.