Representating groups on graphs

Details Representating groups on graphs Representating groups on graphs

2009-04-24

arxiv.org/abs/0904.3941v1

Computer Science

Computational Complexity

13 pages, 2 figures

Scientific paper

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.

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