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 $0<\frac{c}{d}\le 1$. A $\frac{c}{d}$-dominating function (partial signed dominating function) is a function $f:V\to \{-1,1\}$ such that $f(N[v])\ge c$ 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 ${\gamma }_{\frac{c}{d}}(G)$.