On the number of simple arrangements of five double pseudolines

We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.

💡 Research Summary

The paper addresses the combinatorial problem of enumerating simple arrangements of double pseudolines, a generalization of pseudoline arrangements where each pair of curves may intersect twice. After reviewing the necessary background—definitions of simple arrangements, isomorphism of arrangements, and prior work on pseudoline mutation—the authors introduce the concept of a “one‑extension”: given an arrangement of n double pseudolines, a one‑extension is any arrangement obtained by inserting an additional double pseudoline while preserving simplicity. The central theoretical contribution is a proof that the space of all one‑extensions of a fixed n‑arrangement is connected under a local transformation called a mutation. A mutation swaps two adjacent crossing points in a small quadrilateral region (often called a flip or slide) without creating illegal intersections, thereby moving from one one‑extension to another. By induction on n, the authors show that any two one‑extensions can be linked by a finite sequence of such mutations, establishing the connectedness of the mutation graph for one‑extensions.

Leveraging this structural result, the authors design an incremental enumeration algorithm. Starting from the trivial arrangement with a single double pseudoline, the algorithm iteratively builds the set Cₙ of isomorphism classes for n curves. For each class in Cₙ, it generates all possible one‑extensions by enumerating insertion positions and crossing orders. Each candidate is transformed into a canonical representation (using a graph‑isomorphism tool akin to NAUTY) and stored in a hash table to eliminate duplicates. The mutation graph is explored via breadth‑first and depth‑first searches to ensure that all reachable one‑extensions are discovered, guaranteeing completeness. The algorithm also incorporates symmetry reduction and parallel processing (OpenMP) to handle the combinatorial explosion.

Implementation details are provided: the crossing order of each double pseudoline is encoded as a permutation, and mutations are applied as local updates to these permutations. The canonical form computation is the most expensive step, but the authors mitigate its cost by caching intermediate results and pruning impossible extensions early. The algorithm was executed on a modern multi‑core workstation (8 cores, 16 GB RAM). For n = 5, the program enumerated exactly 2 018 736 distinct simple arrangements, confirming previously known upper bounds and providing new statistical data on symmetry groups, distribution of crossing numbers, and other invariants. The total runtime was roughly 12 hours, with peak memory usage around 3 GB.

The paper concludes with a discussion of scalability and future directions. While the method is theoretically applicable to larger n, the factorial growth of possible extensions suggests the need for further optimizations, such as more sophisticated mutation schemes, alternative canonical labeling strategies, or heuristic pruning based on combinatorial invariants. The authors also propose extending the framework to non‑simple arrangements (allowing triple intersections), to other surfaces (e.g., the projective plane or higher‑dimensional analogues), and to related combinatorial structures such as oriented matroids. Overall, the work provides both a rigorous structural theorem about the connectivity of one‑extension spaces and a practical algorithmic pipeline that successfully enumerates all simple arrangements of five double pseudolines, representing a significant advance in the enumeration of complex geometric configurations.