Fault-tolerant strong Menger connectivity with conditional faults on wheel networks

An interconnection network is usually modeled as a graph, where vertex and edge correspond the processor and the link between two distinct processors, respectively. Connectivity is an important metric for the fault tolerance in interconnection networks. A connected graph $G$ is called strongly Menger connected if each pair of vertices $u$ and $v$ are connected by $min\{d_G\left(u\right),d_G\left(v\right)\}$ vertex-disjoint paths in $G$. A graph $G$ is called $m$-fault-tolerant strongly Menger ($m$-FTSM for short) connected if $G-F$ remains strongly Menger connected for any $F\subseteq V\left(G\right)$ with $\left|F\right|\le m$. A graph $G$ is called $m$-conditional fault-tolerant strongly Menger ($m$-CFTSM for short) connected if $G-F$ remains strongly Menger connected for any $F\subseteq V\left(G\right)$ with $\left|F\right|\le m$ and $\delta (G-F)\ge 2$. In this paper, we show that $n$-dimensional wheel network $C{W}_n$ is $(2n-4)$-FTSM connected for $n\ge 4$, $(2n-5)$-fault-tolerant one-to-many strongly Menger connected for $n\ge 4$, and $(4n-10)$-CFTSM connected for $n\ge 5$. Moreover, the bounds $(2n-4)$, $(2n-5)$ and $(4n-10)$ are sharps.