Generalized 4-connectivity of hierarchical star networks

Abstract The connectivity is an important measurement for the fault-tolerance of a network. generalized natural generalization classical connectivity. An S S -tree connected graph G G tree T = ( V ? , E ) T=\left(V^{\prime} ,E^{\prime} ) that contains all vertices in subject to ? S\subseteq V\left(G) . Two -trees T and T^{\prime} are internally disjoint if only ? ? E\left(T)\cap E\left(T^{\prime} )=\varnothing V\left(T)\cap V\left(T^{\prime} )=S Denote by ? \kappa \left(S) maximum number k k -connectivity defined as min { ? width="0.1em" and } {\kappa }_{k}\left(G)=\min \left\{\kappa \left(S)| V\left(G)\hspace{0.33em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{0.33em}| S| \hspace{0.33em}=\hspace{0.33em}k\right\} Clearly, 2 }_{2}\left(G)=\kappa \left(G) In this article, we show 4 H n ? 1 }_{4}\left(H{S}_{n})=n-1 , where H{S}_{n} hierarchical star