Representating groups on graphs

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.

💡 Research Summary

The paper introduces and studies the group representability problem on graphs. Given a finite group G and a finite undirected graph X, the task is to decide whether there exists a non‑trivial homomorphism φ : G → Aut(X), i.e., a way to “represent” the elements of G as automorphisms of X. This formulation naturally bridges group theory and graph isomorphism, prompting the authors to explore the computational complexity of the problem under various restrictions on G and on the host graph.

Main result for abelian and solvable groups.
The authors first consider groups that are either abelian or belong to the class of solvable groups. They show that, even when the group is supplied as a permutation group via a generating set, the representability problem is polynomial‑time Turing equivalent to the classic Graph Isomorphism (GI) problem. The reduction works in both directions.

• From GI to representability: given two graphs Y₁ and Y₂, one constructs a single graph X that encodes the structure of a chosen abelian (or solvable) group A together with Y₁ and Y₂. Then A is representable on X if and only if Y₁ ≅ Y₂.
• From representability to GI: given a group G (abelian or solvable) and a graph X, one builds a pair of auxiliary graphs whose isomorphism precisely captures the existence of a non‑trivial homomorphism G → Aut(X). Both constructions run in polynomial time. Consequently, the decision problem for abelian‑group representability, solvable‑group representability, and GI all lie in the same complexity class; any breakthrough for one immediately transfers to the others.

General groups on trees.
The second part of the paper turns to the case where the host graph is a tree. Although tree isomorphism is known to be solvable in linear time, the representability problem becomes dramatically harder. The authors prove that, for an arbitrary finite group G and an integer n, deciding whether there exists a non‑trivial homomorphism G → Sₙ (the symmetric group on n points) is log‑space equivalent to asking whether G is representable on some tree with n leaves. The reduction is straightforward: a star‑shaped tree with n pendant vertices has an automorphism group isomorphic to Sₙ; thus a representation of G on this tree exists exactly when G admits a non‑trivial permutation representation of degree n. Conversely, any representation of G on a tree can be transformed into a permutation representation of comparable degree.

The existence of a non‑trivial homomorphism G → Sₙ is a classic problem in computational group theory, for which no polynomial‑time algorithm is known in the general case. It subsumes the permutation representation problem and is believed to be at least as hard as Graph Isomorphism, possibly harder (e.g., NP‑hard for certain families of groups). Hence, despite the simplicity of trees, the group‑representability question on trees inherits this difficulty, highlighting a striking contrast with the tractability of plain tree isomorphism.

Implications and future directions.
The paper’s findings have several noteworthy consequences:

1. Complexity classification. For abelian and solvable groups, representability does not introduce a new complexity class; it is essentially another incarnation of GI. This suggests that any algorithmic advances for GI (e.g., quasipolynomial‑time algorithms) automatically apply to these group‑representability instances.

2. Hardness beyond GI. The reduction to the permutation‑representation problem shows that, when the host graph is a tree, the problem can be strictly harder than GI. This opens a new line of inquiry into which graph families preserve the “GI‑equivalence” and which amplify the difficulty.

3. Algorithmic prospects. The equivalence with GI invites the use of existing GI solvers (e.g., Babai’s quasipolynomial algorithm) for abelian/solvable representability. For the tree case, one may explore parameterized algorithms (e.g., fixed‑parameter tractable with respect to the degree n or the composition length of G) or approximation schemes.

4. Broader connections. The work links group representation theory, graph automorphism groups, and classic isomorphism problems. It suggests that other algebraic representation questions (e.g., representing rings or modules on graphs) might exhibit similar complexity patterns.

In summary, the authors formulate a natural decision problem at the intersection of group theory and graph theory, demonstrate that for abelian and solvable groups the problem is computationally equivalent to Graph Isomorphism, and reveal that for arbitrary groups on trees the problem collapses to the existence of non‑trivial permutation representations—a problem for which no polynomial‑time algorithm is currently known. These results deepen our understanding of how algebraic structure interacts with graph‑theoretic symmetry and point to rich avenues for further research in both theoretical computer science and computational algebra.