Representating groups on graphs

TL;DR

This paper explores the computational complexity of representing various groups on graphs and trees, revealing equivalences to graph isomorphism problems and highlighting the difficulty of certain group homomorphism checks.

Contribution

It establishes the complexity equivalences between group representability on graphs and graph isomorphism, and characterizes the problem for groups on trees.

Findings

Abelian and solvable group representability are equivalent to graph isomorphism.

Representability on trees relates to homomorphism existence checks.

No known polynomial time algorithms for general group representability on trees.

Abstract

In this paper we formulate and study the problem of representing groups on graphs. We show that with respect to polynomial time turing reducibility, both abelian and solvable group representability are all equivalent to graph isomorphism, even when the group is presented as a permutation group via generators. On the other hand, the representability problem for general groups on trees is equivalent to checking, given a group $G$ and $n$, whether a nontrivial homomorphism from $G$ to $S_n$ exists. There does not seem to be a polynomial time algorithm for this problem, in spite of the fact that tree isomorphism has polynomial time algorithm.