On the Lower Bounds of Partial Signed Domination Number of Graphs

Abstract

Let \(G = (V, E)\) be a simple graph. For any real-valued function \(f: V \to {R}\) and \(S \subseteq V\), let \(f(S) = \sum_{v \in S} f(v)\). Let \(c, d\) be positive integers such that \(\gcd(c, d) = 1\) and \(0 < \frac{c}{d} \leq 1\). A \(\frac{c}{d}\)-dominating function (partial signed dominating function) is a function \(f: V \to \{-1, 1\}\) such that \(f(N[v]) \geq c\) for at least \(c\) of the vertices \(v \in V\). The \(\frac{c}{d}\)-domination number (partial signed domination number) of \(G\) is \(\gamma_{\frac{c}{d}}(G) = \min \{f(V) | f \text{ is a } \frac{c}{d}\text{-dominating function on } G\}\). In this paper, we obtain a few lower bounds of \(\gamma_{\frac{c}{d}}(G)\).