Hyper K1,r and sub-K1,r fault tolerance of star graphs

Given a connected subgraph H of an interconnection network G, the H-structure connectivity & kappa;(G; H) and the H-substructure connectivity & kappa;s(G; H) of G are two important measurements for fault tolerance of G. The n-dimensional star graph Sn is an attractive interconnection network candidate for multiprocessor systems, and it is (n - 1)-regular. Let n & GE; 4 and 0 < r < n - 1. In this paper, we prove that (i) & kappa;(Sn; K1,r) = & kappa;s(Sn; K1,r) = n - 1, and (ii) the removal of any minimum K1,r-cut (or, minimum sub-K1,r-cut) will split Sn into exactly two components, one of which is a singleton, where K1,r is a tree on 1+ r vertices, r of which are leaves. The main results generalize some known ones.& COPY; 2023 Elsevier B.V. All rights reserved.