Fault tolerant supergraphs with automorphisms

Abstract

Given a graph Y on n vertices and a desired level of fault-tolerance k, an objective in fault-tolerant system design is to construct a supergraph X on n+k vertices such that the removal of any k nodes from X leaves a graph containing Y. In order to reconfigure around faults when they occur, it is also required that any two subsets of k nodes of X are in the same orbit of the action of its automorphism group. In this paper, we prove that such a supergraph must be the complete graph. This implies that it is very expensive to have an interconnection network which is k-fault-tolerant and which also supports automorphic reconfiguration. Our work resolves an open problem in the literature. The proof uses a result due to Cameron on k-homogeneous groups.