Component connectivity of the hypercubes

Component connectivity of generalized Petersen graphs

“…Notice that κ r (G) is defined as the minimum number of vertices that must be removed from G in order to obtain a graph with at least r connected components, since for some graphs G it is not possible to obtain a graph with exactly r connected components by removing vertices. This concept was originally introduced by Sampathkumar [6] and has been recently studied for hypercubes by Hsu et al [5]. It was shown in [6] that κ r (G) ≤ λ r (G) for r = 2, .…”

“…n − 1. These graphs were introduced by Coxeter [3] and named by Watkins [9] based on the fact that GP (5,2) is the Petersen graph. Different equivalent definitions of generalized Petersen graphs can be found in the literature.…”

Conditional fault diagnosis of hierarchical hypercubes

“…Recently, Yang and Meng [27] determined the extra connectivity of hypercubes, Hsu et al [12] went further to establish the component connectivity of hypercubes. Yang et al [31], Cheng and Lipman [2] and Cheng and Lipták [3] evaluated the size of surviving graph S n − F of star graph S n , where F is a subset of V (S n ) with |F| ≤ 2n − 4.…”

The 4-component connectivity of alternating group networks

“…For an integer 2, the generalized -connectivity of a graph G, denoted by κ (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least components or a graph with fewer than vertices. A graph G is (n, )-connected if κ (G) n. A synonym for such a generalization was also called the general connectivity by Sampathkumar [26] or -component connectivity ( -connectivity for short) by Hsu et al [18], Cheng et al [7][8][9] and Zhao et al [29]. Hereafter, we follow the use of the terminology of Hsu et al Obviously, κ 2 (G) = κ(G).…”

“…However, determining -connectivity is still unsolved for most interconnection networks. As a matter of fact, it has been pointed out in [18] that, unlike the hypercube, the results of the well-known interconnection networks such as the star graphs [1] and the alternating group graphs [20] are still unknown.…”