Counting Triangulations of Fixed Cardinal Degrees

2025-10-06

AI Summary

This paper studies the enumeration of straight-line triangulations of a fixed planar point set, subject to prescribed orthogonal degree sequences—i.e., the number of neighbors in each of the four cardinal directions (north, south, east, west) at every vertex. We show that such local directional constraints are insufficient to uniquely determine the triangulation; moreover, counting triangulations satisfying given orthogonal degrees is #P-hard. This constitutes the first rigorous characterization of both uniqueness and computational complexity for triangulations under orthogonal degree constraints. We establish #P-hardness via a polynomial-time reduction from the #3-regular bipartite planar vertex cover problem to our triangulation counting problem. Our results demonstrate that even with fine-grained local directional information, planar embedded combinatorial structures inherently admit ambiguity, and their enumeration remains computationally intractable.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if we know, for each vertex, the number of neighbors in each of the four cardinal directions, the triangulation is not completely determined. In fact, we show that counting such triangulations is #P-hard via a reduction from #3-regular bipartite planar vertex cover.

Problem

• Counting triangulations with fixed vertex cardinal degrees
• Determining uniqueness from directional neighbor constraints
• Proving #P-hardness via planar graph reduction

Innovation

• Counting triangulations with fixed cardinal degrees
• Using #P-hardness proof via reduction
• Analyzing planar straight-line triangulation constraints