Generalized measures of fault tolerance in hypercube networks

A vertex subset F is an R g -cut of a connected graph G if G − F is disconnected and every vertex in G − F has at least g fault-free neighbors in G − F . The cardinality of the minimum R g -cut of G is the R g -connectivity of G , denoted by κ g ( G ) . This parameter measures a kind of conditional fault tolerance of networks. In this work, we characterize the smallest components after deleting a minimum R g -cut of hypercubes. Our work strengthens the results of [A.H. Esfahanian, Generalized measure of fault tolerance with application to N -cube networks, IEEE Trans. Comput. 38 (1989) 1586–1591] and [S. Latifi, M. Hegde, M. Naraghi-Pour, Conditional connectivity measures for large multiprocessor systems, IEEE Trans. Comput. 43 (1994) 218–222] and also corrects bugs in them.