Generalized 4-connectivity of hierarchical star networks

Abstract

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