Mei-Mei Gu

The pessimistic diagnosabilities of some general regular graphs

The pessimistic diagnosis strategy is a classic strategy based on the PMC model. A system is t / t -diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by t p ( G ) , is the maximal number of faulty processors so that the system G is t / t -diagnosable. In this paper, we study the pessimistic diagnosabilities of some general k-regular k-connected graphs G n . The main result t p ( G n ) = 2 k - 2 - g under some conditions is obtained, where g is the maximum number of common neighbors between any two adjacent vertices in G n . As applications of the main result, every pessimistic diagnosability of many famous networks including some known results, such as the alternating group networks AN n , the k-ary n-cubes Q n k , the star graphs S n , the matching composition networks G ( G 1 , G 2 ; M ) and the alternating group graphs AG n , are obtained. We study the pessimistic diagnosabilities of some general regular graphs under the PMC model.The exact value of t p ( G n ) is obtained.As applications, the pessimistic diagnosability of AN n , Q n k , S n , some MCN and AG n etc., are obtained.

The g-good-neighbor diagnosability of (n,k)-star graphs

Many large-scale multiprocessor or multi-computer systems take interconnection networks as underlying topologies. Fault diagnosis is especially important to identify fault tolerability of such systems. The g-good-neighbor (conditional) diagnosability such that every fault-free node has at least g fault-free neighbors is a novel measure of diagnosability. In this paper, we show that the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the PMC model ( 2 ź k ź n - 1 and 1 ź g ź n - k ) and the comparison model ( 2 ź k ź n - 1 and 2 ź g ź n - k ) is n + g ( k - 1 ) - 1 , respectively. In addition, we derive that 1-good-neighbor diagnosability of S n , k under the comparison model is n + k - 2 for 3 ź k ź n - 1 and n ź 4 . As a supplement, we also derive that the g-good-neighbor diagnosability of the ( n , 1 ) -star graph S n , 1 ( 1 ź g ź ź n / 2 ź - 1 and n ź 4 ) under the PMC model and the comparison model is ź n / 2 ź - 1 , respectively. We explore fault diagnosability of multiprocessor systems based on combinatorial network theory.We establish the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the PMC model.We establish the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the comparison model.

Fault-Tolerant Cycle Embedding in Balanced Hypercubes with Faulty Vertices and Faulty Edges

Let Fv (resp. Fe) be the set of faulty vertices (resp. faulty edges) in the n-dimensional balanced hypercube BHn. The edge-bipancyclicity of BHn − Fv for |Fv| ≤ n − 1 had been proved in [Inform. Sci. 288 (2014) 449–461]. The existence of edge-Hamiltonian cycles in BHn − Fe for |Fe| ≤ 2n − 2 were obtained in [Appl. Math. Comput. 244 (2014) 447–456]. In this paper, we consider fault-tolerant cycle embedding of BHn with both faulty vertices and faulty edges, and prove that there exists a fault-free cycle of length 22n − 2|Fv| in BHn with |Fv| + |Fe| ≤ 2n − 3 and |Fv| ≤ n − 1 for n ≥ 2.

The pessimistic diagnosability of bubble-sort star graphs and augmented k-ary n-cubes

ABSTRACT A system is t/t-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by , is the maximal number of faulty processors so that the system G is t/t-diagnosable. The known results about for alternating group graphs [Inform. Process. Lett. (2015), pp. 151–154]; BC networks [IEEE Trans. Comput. (2005), pp. 176–184]; the k-ary n-cube networks [IEEE Trans. Comput. (1991), pp. 232–237], [Int. J. Comput. Math. (2012), pp. 1–10] etc. have property that . In this paper, we study the pessimistic diagnosability of two kinds of graphs with , those are: bubble-sort star graphs and augmented k-ary n-cubes , and prove that for , for , and for and .

Conditional fault-tolerant edge-bipancyclicity of hypercubes with faulty vertices and edges

Let F be a faulty set in an n-dimensional hypercube Q n such that in Q n - F each vertex is incident to at least two edges, and let f v , f e be the numbers of faulty vertices and faulty edges in F, respectively. In this paper, we consider the fault-tolerant edge-bipancyclicity of hypercubes. It is shown that each edge in Q n - F for n ? 3 lies on a fault-free cycle of any even length from 6 to 2 n - 2 f v if f v + f e ? 2 n - 5 . This gives an answer for a problem proposed by Yang et al. (2016) 33. Consider hypercubes with mixed faulty edges and faulty vertices under conditional-fault model.Improve the previous results about the fault-tolerant edge-bipancyclicity of hypercubes.Give an affirmative answer for a problem proposed by Yang et al. (2016).

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 systems 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 n 2 ) = 4 n − 9 for split-star networks S n 2 ; t p ( Γ n ) = 2 n − 4 for Cayley graphs generated by transposition trees Γ n ; t p ( Γ n ( Δ ) ) = 4 n − 11 for Cayley graph generated by the 2-tree Γ n ( Δ ) ; t p ( B P n ) = 2 n − 2 for the burnt pancake networks B P 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.

The 1-Good-Neighbor Conditional Diagnosability of Some Regular Graphs

The g-good-neighbor conditional diagnosability is the maximum number of faulty vertices a network can guarantee to identify, under the condition that every fault-free vertex has at least g fault-free neighbors. In this paper, we study the 1-good-neighbor conditional diagnosabilities of some general k-regular k-connected graphs G under the PMC model and the MM* model. The main result t1(G)=2k−l−1 under some conditions is obtained, where l is the maximum number of common neighbors between any two adjacent vertices in G. Moreover, the following results are derived: t1(HSn)=2n−1 for the hierarchical star networks, t1(Xn)=2n−1 for the BC networks, t1(AGn)=4n−10 for the alternating group graphs AGn.

Edge Fault-Tolerant Strong Hamiltonian Laceability of Balanced Hypercubes

The balanced hypercube BHn, proposed by Wu and Huang, is a new variation of hypercube. A Hamiltonian bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path between two arbitrary vertices from different partite sets. A Hamiltonian laceable graph G is strongly Hamiltonian laceable if there is a path of length | V(G) | − 2 between any two distinct vertices of the same partite set. A graph G is called k-edge-fault strong Hamiltonian laceable, if G – F is strong Hamiltonian laceable for any edge-fault set F with | F | ≤ k. It has been proved that the balanced hypercube BHn is strong Hamiltonian laceable. In this paper, we improve the above result and prove that BHn is (n – 1)-edge-fault strong Hamiltonian laceable.

A Note on the Pessimistic Diagnosability of Augmented Cubes

A system is t/t-diagnosable if, provided the number of faulty processor is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistake as a faulty one. The pessimistic diagnosability of a system G, denoted by tp(G), is the maximal number of faulty processors so that the system G is t/t-diagnosable. The augmented cube AQn, proposed by Choudum and Sunitha [Networks 40 (2) (2002) 71–84], has many attractive properties such as regularity, strong connectivity and symmetry. In this paper, we determine the pessimistic diagnosability of the n-dimensional augmented cube AQn and prove that tp(AQn) = 4n− 8 for n ≥ 5.