Graph Isomorphism and Circuit Size

TL;DR

This paper explores the relationship between Graph Automorphism and Rigid Graph Isomorphism, showing that the reduction can be achieved with a specific type of reduction, highlighting nuances in graph isomorphism complexity.

Contribution

It demonstrates that the reduction from Graph Automorphism to Rigid Graph Isomorphism can be performed using Grollman and Selman's 'smart reduction' method.

Findings

Reduction achievable with 'smart reduction'

Clarifies complexity relationship between automorphism and isomorphism

Highlights nuances in graph isomorphism problems

Abstract

It is well-known [KST93] that the complexity of the Graph Automorphism problem is characterized by a special case of Graph Isomorphism, where the input graphs satisfy the "promise" of being rigid (that is, having no nontrivial automorphisms). In this brief note, we observe that the reduction of Graph Automorphism to the Rigid Graph Ismorphism problem can be accomplished even using Grollman and Selman's notion of a "smart reduction".