A Kind Of Conditional Vertex Connectivity Of Cayley Graphs Generated By Wheel Graphs. (13th November 2019) - Luo, Zuwen; Xu, Liqiong
Abstract: Let $G=(V(G), E(G))$ be a connected graph. A subset $T \subseteq V(G)$ is called an $R^{k}$ -vertex-cut, if $G-T$ is disconnected and each vertex in $V(G)-T$ has at least $k$ neighbors in $G-T$ . The cardinality of a minimum $R^{k}$ -vertex-cut is the $R^{k}$ -vertex-connectivity of $G$ and is denoted by $κ ^{k}(G)$ . $R^{k}$ -vertex-connectivity is a new measure to study the fault tolerance of network structures beyond connectivity. In this paper, we study $R^{1}$ -vertex-connectivity and $R^{2}$ -vertex-connectivity of Cayley graphs generated by wheel graphs, which are denoted by $AW_{n}$, and show that $κ ^{1}(AW_{n})=4n-7$ for $n≥ 6$ ; $κ ^{2}(AW_{n})=6n-12$ for $n≥ 6$ .