Conditional (edge) neighbor connectivity of hierarchical star networks

Neighbor connectivity can be utilized to assess fault-tolerance and robustness of interconnection networks, even though it was first proposed to measure the degree of damage in espionage networks brought on by the subterranean rebellion movement. We introduce a novel parameter—conditional (edge) neighbor connectivity, an improvement of neighbor connectivity by requiring that any component of the survival graph is nontrivial. Let G be a network, the conditional neighbor connectivity κNB′(G) (resp., conditional edge neighbor connectivity λNB′(G)) is characterized by the smallest amount of vertices (resp., edges) such that the elimination of the closed neighborhoods of these vertices (resp., edges) results in a nontrivial complete graph or disconnected network with no trivial component (as for edge version, the resulting network only needs to satisfy the second condition). This work explores the conditional (edge) neighbor connectivity of hierarchical star network HSn, which uses the star graph Sn as underlying module. In detail, we acquire κNB′(HSn)=λNB′(HSn)=2n−3 for n≥5.