Signed Total \(k\)-Domination in Graphs

Abstract

A signed total \(k\)-dominating function of a graph \(G = (V, E)\) is a function \(f: V \rightarrow \{+1, -1\}\) 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 \(g\), \(f \neq g\), for which \(g(v) \leq f(v)\) 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 \(\Gamma_{t,k}^s(G)\) 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 \(\Gamma_{t,k}^s(G)\) for an arbitrary graph.