Component connectivity of wheel networks

The r-component connectivity cκ_r(G) 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, cκ_r(G) is reduced to the classic notion of connectivity κ(G). So cκ_r(G) is a generalization of κ(G), and is therefore a more general and more precise measurement for the reliability of large interconnection networks. The m-dimensional wheel network CW_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 cκ₃(CW_m)=4m−7 for m≥5, cκ₄(CW_m)=6m−13 and cκ₅(CW_m)=8m−20 for m≥6.