  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.

Discrete Mathematics;Networking and Internet Architecture;Combinatorics