Equal relation between the extra connectivity and pessimistic diagnosability for some regular graphs

Abstract

Extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processor. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty vertices within a set containing at most one fault-free vertex. In this paper, the result that the pessimistic diagnosability t p (G) equals the extra connectivity κ 1 (G) of a regular graph G under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability t p (S 2 n ) = 4n − 9 for split-star networks S 2 n ; t p (Γ n ) = 2n − 4 for Cayley graphs generated by transposition trees Γ n ; t p (Γ n (∆)) = 4n−11 for Cayley graph generated by the 2-tree Γ n (∆); t p (BP n ) = 2n−2 for the burnt pancake networks BP n . As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs; the alternating group networks; BC networks; the k-ary n-cube networks etc. are obtained directly.

Structure Connectivity and Substructure Connectivity of $k$ -Ary $n$ -Cube Networks

“…One crucial parameter to evaluate the fault tolerability of a network is its connectivity. The connectivity of a graph G, denoted by κ(G), is the minimum cardinality of a node set F ⊆ V (G), such that F's deletion disconnects G. As variants of the classic node-connectivity, several kinds of conditional connectivity were proposed and studied [2], [3], [5], [6], [8], [9], [12]- [16], [18], [23], [25], [26]. Notably among them, Fàbrega and Fiol [4] introduced the g-extra connectivity.…”

“…Obviously, Q k 1 is a cycle of length k, Q 2 n is an n-dimensional hypercube. Q 6 1 and Q 4 2 are depicted in FIGURE 2. Two distinct adjacent nodes are neighbours.…”

Measuring the Vulnerability of Alternating Group Graphs and Split-Star Networks in Terms of Component Connectivity

“…In particular, Cheng et al [14] showed that alternating group graphs and split-stars are superior to the n-cubes and star graphs under the comparison using an advanced vulnerability measure called toughness, which was defined in [22]. For the two families of graphs, many researchers were attracted to study fault tolerant routing [12], fault tolerant embedding [5], [6], [42], matching preclusion [2], [11], restricted connectivity [15], [25], [35], [36], [48] and diagnosability [10], [25], [30], [34]- [36], [41]. Moreover, alternating group graphs are also edge-transitive and possess stronger and rich properties on Hamiltonicity (e.g., it has been shown to be not only pancyclic and Hamiltonian-connected [33] but also panconnected [6], panpositionable [40] and mutually independent Hamiltonian [39]).…”

“…Given a graph G and a nonnegative integer h, the h-extra connectivity of G, denoted by κ (h) (G), is the cardinality of a minimum vertex-cut S of G, if it exists, such that each component of G − S has at least h + 1 vertices. In fact, the extra connectivity plays an important indicator of a network's ability for diagnosis and fault tolerance [25], [31], [35], [36]. Currently, the known results of h-extra connectivity for alternating group graphs and splitstars were proposed in [36] and [35], respectively.…”

The Component Connectivity of Alternating Group Graphs and Split-Stars

“…In particular, Cheng et al [12] showed that alternating group graphs and split-stars are superior to the n-cubes and star graphs under the comparison using an advanced vulnerability measure called toughness, which was defined in [20]. For the two families of graphs, many researchers were attracted to study fault tolerant routing [10], fault tolerant embedding [5,6,37], matching preclusion [2,9], restricted connectivity [13,22,30,31,39] and diagnosability [8,22,25,[29][30][31]36]. Moreover, alternating group graphs are also edge-transitive and possess stronger and rich properties on Hamiltonicity (e.g., it has been shown to be not only pancyclic and Hamiltonian-connected [28] but also panconnected [6], panpositionable [35] and mutually independent Hamiltonian [34]).…”