# Fault Tolerance in Cayley Graphs

This report mainly focuses on the fault tolerance of networks (graphs), which is measured in two ways. The first is preclusion, which is a vertex set, whose removal leads to the destruction of all the subgraphs of specific size in the hierarchical network and the preclusion number is the minimum of this vertex set. Here we investigate a class of graphs which are constructed by combining the star graph with the bubble-sort graph, and give some preclusion numbers and edge preclusion numbers for this class of graphs. The second is $g$-good-neighbor diagnosability, which is the cardinality of a faulty vertex set, whose removal leads to that each vertex has at least $g$ fault-free neighbors in the remaining graph. Here we prove the $1$-good-neighbor diagnosability of $CK_n$ under the PMC model is $n^2-n-1$ for $n\geq 4$ and under the MM$^*$ model is $n^2-n-1$ for $n\geq 5$.