Generalized 4-connectivity of hierarchical star networks
Junzhen Wang, Jinyu Zou, Shumin Zhang · Open Mathematics · 2022
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