Component connectivity of wheel networks

The r-component connectivity $c{\kappa }_r\left(G\right)$ of a noncomplete graph G is the size of a minimum set of vertices, whose deletion disconnects G such that the remaining graph has at least r components. When $r=2$, $c{\kappa }_r\left(G\right)$ is reduced to the classic notion of connectivity $\kappa \left(G\right)$. So $c{\kappa }_r\left(G\right)$ is a generalization of $\kappa \left(G\right)$, and is therefore a more general and more precise measurement for the reliability of large interconnection networks. The m-dimensional wheel network $C{W}_m$ was first proposed by Shi and Lu in 2008 as a potential model for the interconnection network [19], and has been getting increasing attention recently. It belongs to the category of Cayley graphs, and possesses some properties desirable for interconnection networks. In this paper, we determine the r-component connectivity of the wheel network for $r=3,4,5$. We prove that $c{\kappa }_3\left(C{W}_m\right)=4m-7$ for $m\ge 5$, $c{\kappa }_4\left(C{W}_m\right)=6m-13$ and $c{\kappa }_5\left(C{W}_m\right)=8m-20$ for $m\ge 6$.