Signed Total $k$-Domination in Graphs

Abstract

A signed total $k$-dominating function of a graph $G=(V,E)$ is a function $f:V\to \{+1,-1\}$ such that for every vertex $v$, the sum of the values of $f$ over the open neighborhood of $v$ is at least $k$. A signed total $k$-dominating function $f$ is minimal if there does not exist a signed total $k$-dominating function $g$, $f\ne g$, for which $g(v)\le f(v)$ for every $v\in V$.The weight of a signed total $k$-dominating function is $w(f)=\sum _{v\in V}f(v)$. The signed total $k$-domination number of $G$, denoted by ${\gamma }_{t,k}^s(G)$, is the minimum weight of a signed total $k$-dominating function on $G$.The upper signed total $k$-domination number ${\mathrm{\Gamma }}_{t,k}^s(G)$ of $G$ is the maximum weight of a minimal signed total $k$-dominating function on $G$.
In this paper, we present sharp lower bounds on ${\gamma }_{t,k}^s(G)$ for general graphs and $K_{r+1}$-free graphs and characterize the extremal graphs attaining some lower bounds. Also, we give a sharp upper bound on ${\mathrm{\Gamma }}_{t,k}^s(G)$ for an arbitrary graph.