The Isomorphism Problem for Planar 3-Connected Graphs is in Unambiguous Logspace

The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3-connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class $AC^1$. In this paper we improve the upper bound for planar 3-connected graphs to unambiguous logspace, in fact to $UL \cap coUL$. As a consequence of our method we get that the isomorphism problem for oriented graphs is in $NL$. We also show that the problems are hard for $L$.

💡 Research Summary

The paper addresses the graph isomorphism problem (GI) for planar 3‑connected graphs, improving the known upper bound from AC¹ to unambiguous logspace (UL) intersect co‑UL. The authors follow the classic approach of constructing a canonical code for each graph, but they redesign every step so that it can be executed using only logarithmic space, and in a way that guarantees at most one accepting computation (unambiguity).

The algorithm consists of three main phases. First, given a planar embedding represented by a rotation scheme ρ and a designated directed edge (s, t), the algorithm builds a canonical spanning tree T of the graph. Instead of a standard breadth‑first or depth‑first search, which would require storing visited vertices, the construction uses distances between vertices. Distances are computed by a recent result that reachability (or “re‑usability”) in planar directed graphs lies in UL ∩ co‑UL. With these distances, the algorithm walks from the root s towards any vertex w, always choosing the first edge (according to ρ) that lies on a shortest s‑w path. This yields a unique tree that depends only on ρ, (s, t) and the edge set, not on the particular representation.

Second, using the canonical tree T, the algorithm produces a canonical list L of all undirected edges. Starting from (s, t) as the active edge, it traverses the graph in a depth‑first manner: if the active edge belongs to T, it follows the tree edge and then switches to the successor edge around the opposite endpoint according to ρ; otherwise it proceeds by rotating around the current endpoint using ρ. This process visits each undirected edge exactly twice (once in each direction) and requires only the current active edge as memory, thus fitting into deterministic logspace (L).

Third, the list L is transformed into a canonical code by renaming vertices according to their first appearance order. The algorithm scans L left‑to‑right, assigning the integer 1 to the first vertex encountered, 2 to the next new vertex, and so on. Whenever a vertex reappears, the previously assigned number is output. This renaming can be performed in logspace by keeping a counter and, for each vertex, locating its first occurrence to retrieve its number.

The final code, denoted code(G, ρ, s, t), is invariant under graph isomorphism: two planar 3‑connected graphs G and H are isomorphic iff there exists a choice of designated edge and rotation scheme for H that yields the same code as G. To test isomorphism, the algorithm fixes (s, t) and ρ for G, then iterates over all edges of H and both possible rotation schemes (ρ and its inverse), generating the corresponding codes and checking for equality. The outer loop uses only O(log n) space, while each inner iteration invokes the three phases already shown to be in UL ∩ co‑UL. Consequently, planar 3‑connected GI belongs to UL ∩ co‑UL.

The paper also shows that when the rotation scheme is part of the input (i.e., for oriented graphs), the same construction yields an NL algorithm, because the unambiguity requirement disappears. Moreover, the authors prove that planar 3‑connected GI is L‑hard, establishing that the problem is unlikely to be solvable in deterministic logspace.

Key technical contributions include: (1) adapting the Bourke‑Tewari‑Vinodchandran result on planar directed reachability to compute distances in UL ∩ co‑UL; (2) designing a canonical spanning‑tree construction that relies solely on distances and the rotation scheme; (3) showing that the subsequent list generation and renaming steps are logspace‑feasible and canonical. By combining these components, the authors lower the complexity of a classic graph‑theoretic problem from parallel polynomial time to a very low space class, highlighting the power of planar embeddings and rotation schemes in space‑bounded computation.