The m-Component Connectivity of Leaf-Sort Graphs

Connectivity plays an important role in measuring the fault tolerance of interconnection networks. As a special class of connectivity, m-component connectivity is a natural generalization of the traditional connectivity of graphs defined in terms of the minimum vertex cut. Moreover, it is a more advanced metric to assess the fault tolerance of a graph G. Let $G=\left(V\right(G),E(G\left)\right)$ be a non-complete graph. A subset F$(F\subseteq V(G\left)\right)$ is called an m-component cut of G, if $G-F$ is disconnected and has at least m components $(m\ge 2)$. The m-component connectivity of G, denoted by $c{\kappa }_m\left(G\right)$, is the cardinality of the minimum m-component cut. Let $C{F}_n$ denote the n-dimensional leaf-sort graph. Since many structures do not exist in leaf-sort graphs, many of their properties have not been studied. In this paper, we show that $c{\kappa }_3\left(C{F}_n\right)=3n-6$ $(n$ is odd) and $c{\kappa }_3\left(C{F}_n\right)=3n-7$ $(n$ is even) for $n\ge 3$; $c{\kappa }_4\left(C{F}_n\right)=\frac{9n-21}{2}$ $(n$ is odd) and $c{\kappa }_4\left(C{F}_n\right)=\frac{9n-24}{2}$ $(n$ is even) for $n\ge 4$.