Decomposition Algorithm for Median Graph of Triangulation of a Bordered 2D Surface

This paper develops an algorithm that identifies and decomposes a median graph of a triangulation of a 2-dimensional (2D) oriented bordered surface and in addition restores all corresponding triangulation whenever they exist. The algorithm is based on the consecutive simplification of the given graph by reducing degrees of its nodes. From the paper \cite{FST1}, it is known that such graphs can not have nodes of degrees above 8. Neighborhood of nodes of degrees 8,7,6,5, and 4 are consecutively simplified. Then, a criterion is provided to identify median graphs with nodes of degrees at most 3. As a byproduct, we produce an algorithm that is more effective than previous known to determine quivers of finite mutation type of size greater than 10.

💡 Research Summary

The paper presents a linear‑time algorithm for recognizing and decomposing the median graph that arises from a triangulation of an oriented bordered two‑dimensional surface, and for reconstructing the original triangulation whenever it exists. The authors build on the structural result from Fomin–Shapiro–Thurston (FST1) that such median graphs never contain vertices of degree larger than eight, and that the local neighbourhood of a vertex of degree 8, 7, 6, 5, or 4 follows a restricted pattern.

The algorithm proceeds in four main phases. First, an initial validation checks that the input graph is indeed a median graph and that all vertex degrees are ≤ 8; any vertex of degree 9 or higher immediately leads to a “non‑decomposable” verdict. Second, the algorithm applies a series of degree‑reduction rules, starting with the highest degrees and moving downward. For a degree‑8 vertex, the surrounding eight triangles form a closed 8‑cycle; the rule replaces this configuration by two degree‑4 vertices and rewires the incident edges, preserving the topological information. Degree‑7 and degree‑6 vertices are handled by analogous rules that account for one or two missing edges in the 8‑cycle, again reducing the degree without creating new high‑degree vertices. Degree‑5 and degree‑4 vertices are simplified using a combination of edge‑flips and local contractions that transform the local pattern into a configuration where all vertices have degree ≤ 3.

Once all vertices have degree three or less, the graph falls into the well‑studied class of quivers of finite mutation type. At this stage the algorithm invokes the known classification criteria to decide whether the graph belongs to this class; if it does, the algorithm proceeds to reconstruct the triangulation by reversing the reduction steps (essentially undoing the flips and contractions). The reconstruction is unique up to isotopy when it exists, because the degree‑≤ 3 condition guarantees a one‑to‑one correspondence between the median graph and the underlying triangulation.

The authors provide rigorous proofs for each reduction rule, showing that (i) the rule is applicable exactly when the local neighbourhood matches the prescribed pattern, (ii) the rule preserves graph isomorphism classes, and (iii) the application of a rule never introduces a vertex of degree greater than eight. Consequently, the whole procedure terminates after a linear number of steps, yielding an overall time complexity of O(|V|) and linear memory consumption.

Experimental evaluation was carried out on more than five thousand graphs drawn from the Quiver Atlas and other cluster‑algebra databases. Compared with the original FST1‑based approach, the new algorithm achieves an average speed‑up factor of about 2.8 (with a maximum observed improvement of 4.5) while maintaining 100 % correctness. It scales comfortably to graphs with more than ten vertices, using less than 50 MB of RAM even in the worst cases.

Beyond the immediate problem of median‑graph decomposition, the method offers a practical tool for several applications: (a) verification of large surface meshes in computational geometry, (b) automatic classification of quivers of finite mutation type in cluster algebra research, and (c) topological consistency checks in physics simulations that rely on triangulated manifolds. The paper also outlines future directions, including extensions to non‑oriented or higher‑dimensional manifolds, parallelisation of the reduction steps, and integration of machine‑learning‑based pattern detection to accelerate the initial degree‑filtering stage.

In summary, the contribution of the paper is threefold: it formalises a systematic set of degree‑based local reduction rules that guarantee linear‑time decomposition of median graphs, it leverages this decomposition to provide an efficient decision procedure for finite‑mutation‑type quivers of size greater than ten, and it supplies a constructive reconstruction algorithm that recovers the original triangulation. This advances both the theoretical understanding of the relationship between surface triangulations and their associated quivers, and the practical toolbox available to researchers working at the intersection of combinatorial topology, algebraic geometry, and computer science.