Conditional diagnosability of multiprocessor systems based on Cayley graphs generated by transpositions

Abstract

Let Γn be the symmetric group on {1,2,…,n} and S be the generating set of Γn. The corresponding Cayley graph is denoted by Γn(S). If all elements of S are transpositions, a simple way to depict S is to via a graph, called the transposition generating graph of S, denoted by A(S) (or say simply A), where the vertex set of A is {1,2,…,n}, there is an edge in A between i and j if and only if the transposition (ij)∈S, and Γn(S) is called a Cayley graph obtained from a transposition generating graphA. Conditional diagnosability, proposed by Lai et al, is a novel measure of diagnosability that adds the additional condition that any faulty set cannot contain all of the neighbors of any vertex in a system. There are two well-known diagnostic models, PMC model and MM* model. The conditional diagnosability under the PMC (resp. MM*) model of a graph G, denoted by tcPMC(G) (resp. tcMM∗(G)), is the maximum value of t is the maximum value of t such that G is conditionally t-diagnosable under the PMC (resp. MM*) model. The conditional diagnosability of many well-known multiprocessor systems has been explored. In this paper, by exploring the structural properties of these Cayley graphs, suppose |E(A)| is the number of edges in the transposition generating graph A, we obtain the following results:(1) Under the MM* model, tcMM∗(Γn(S))=3|E(A)|−6,if A has a triangle;3|E(A)|−5,if A has no triangles and A is not a star.(2) Under the PMC model, tcPMC(Γn(S))=4|E(A)|−9,if A has a triangle;4|E(A)|−7,if A has no triangles and A is not a star.As corollaries, the conditional diagnosability of many kinds of graphs under the MM* model and the PMC model such as Cayley graphs generated by unicyclic graphs, wheel graphs, complete graphs, tree graphs etc. are obtained.