Fault tolerant supergraphs with automorphisms

TL;DR

This paper proves that constructing a fault-tolerant supergraph with automorphic reconfiguration properties necessarily results in a complete graph, highlighting the high cost of such network designs.

Contribution

It resolves an open problem by showing that the only such supergraph is the complete graph, using Cameron's result on k-homogeneous groups.

Findings

Supergraph must be complete for fault-tolerance and automorphic reconfiguration.

Constructing such supergraphs is very expensive.

The work resolves an open problem in graph theory and network design.

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.