Stability and domination exponentially in some graphs

Abstract

For a graph $G=(V,E)$ and the exponential dominating set $S\subseteq V(G)$ of $G$ such that $\sum_{u \in S}(1/2)^ {\overline{d}(u,v)-1}\geq 1 $, $\forall v\in V(G)$, where $\overline{d}(u,v)$ is the length of a shortest path in $ \langle V(G)-(S-\{u\}) \rangle $ if such a path exists, and $\infty$ otherwise, the minimum exponential domination number, $\gamma_{e}(G)$ is the smallest cardinality of $S$. The minimum exponential domination number can be decreased or increased by removal of some vertices from $G$. In this paper, we continue to study on exponential domination number and stability of some graphs. We consider $\gamma_{e}^{+}$ and $\gamma_{e}^{-}$ stability of the lollipop graph $L_{m,n}$, the comet graph $C_{m,n}$, the sunflower graph $SF_{n}$, the helm graph $H_{n}$, the diamond-necklace graph $N_{n}$, the diamond-bracelet graph $B_{n}$ and the diamond-chain graph $L_{n}$ to give us an idea about the resistance of these graphs.

Keywords

• graph vulnerability
• network design and communication
• domination
• exponential domination number