Fault tolerant supergraphs with automorphisms

Given a basic graph Y and a desired level of fault-tolerance k, an objective in fault-tolerant system design is to construct a supergraph X 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. Index terms — interconnection networks; fault-tolerant system design; structural fault tolerance; graph automorphisms; graph theory; permutation groups.