A Log-space Algorithm for Canonization of Planar Graphs

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell’s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.

💡 Research Summary

The paper establishes that the planar graph isomorphism problem (Planar‑GI) and its stronger variant, planar graph canonization, can both be solved in deterministic logarithmic space (L). This matches the known L‑hardness result (Lindell ’92) and improves the previous upper bound of AC¹ (Miller‑Reif ’91). The authors achieve this by decomposing a planar graph into a hierarchy of components—first into its biconnected components, then each biconnected component into triconnected components—and by applying log‑space algorithms at each level.

The algorithm proceeds in five main stages:

1. Biconnected Component Tree Construction
   Using the log‑space method of Allender, Datta, and Roy (ADK08), the algorithm builds the biconnected component tree of a connected planar graph. Nodes of this tree represent either biconnected components or articulation points. The construction requires only a constant number of pointers and can be performed with a single pass over the adjacency lists.

2. Triconnected Component Decomposition
   For each biconnected component, the algorithm implements a log‑space version of the classic Hopcroft‑Tarjan decomposition (HT73). It identifies all separating pairs (pairs of vertices whose removal disconnects the component) that span a face, classifies them as 3‑connected separating pairs, and replaces each such pair by a virtual edge. By recursively removing separating pairs and adding virtual edges, the component is split into its triconnected pieces (3‑connected graphs, cycles, or 3‑bonds). Lemma 3.4 guarantees that 3‑connected separating pairs that span the same face are non‑intersecting, which allows simultaneous processing in log‑space.

3. Canonization of Triconnected Pieces
   The triconnected components are canonized using the log‑space algorithm of Datta, Limaye, and Nimbhorkar (DLN08). This algorithm exploits planar rotation schemes (embeddings) to produce a unique string (the canon) for each 3‑connected planar graph. The rotation scheme itself can be computed in log‑space (Allender‑Mahajan ’00).

4. Canonization of Biconnected Component Trees
   The authors adapt Lindell’s tree canonization algorithm (Lin92) to the setting where each node of the tree carries the canon of its underlying triconnected component. Direct application of Lindell’s method would require storing the full strings, which exceeds log‑space. To overcome this, the paper introduces two innovations:

   • Compressed Labels: Each node’s canon is replaced by a short hash that can be compared in log‑space while preserving the ordering needed for canonical forms.
   • Group‑Theoretic Lemma (Lemma 5.3): For a coloured 3‑connected planar graph, the size of its automorphism group is bounded by a polynomial in the number of vertices. This bound limits the number of possible label permutations that must be examined, ensuring that the tree‑canonization step remains within log‑space.

5. Global Canonization of the Planar Graph
   Finally, the canon of the whole planar graph is obtained by recursively applying the modified tree canonization to the biconnected component tree, which itself is built from the triconnected component trees. The recursion depth is O(log n) because each decomposition reduces the size of the subproblems, and at each level only a constant amount of information (the current node’s compressed label and a few pointers) needs to be stored.

The paper provides a meticulous complexity analysis. Every subroutine—computing separating pairs, constructing virtual edges, performing reachability queries on the underlying undirected graph, and handling the hash‑based comparisons—uses only O(log n) bits of workspace. The recursion stack also consumes O(log n) bits, yielding an overall deterministic log‑space algorithm.

In addition to the algorithmic contribution, the authors prove a new structural result (Lemma 5.3) that bounds the automorphism group of coloured 3‑connected planar graphs. This result is of independent interest in graph symmetry studies and is essential for the log‑space reduction from biconnected to triconnected cases.

The significance of the work is twofold. First, it settles the exact complexity of planar graph isomorphism: the problem is L‑complete. Second, by providing a log‑space canonization algorithm, it enables memory‑efficient preprocessing for any application that requires a canonical representation of planar structures (e.g., chemical compound comparison, geographic information systems). The techniques introduced—log‑space decomposition of separating pairs, compressed label handling, and the automorphism‑group bound—may inspire similar space‑efficient algorithms for other graph families, such as graphs embeddable on surfaces of bounded genus.

Overall, the paper delivers a comprehensive, rigorously proved log‑space solution to a long‑standing open problem in graph algorithms, advancing both theoretical understanding and practical possibilities for planar graph processing.