Edge-Constrained Hamiltonian Paths on a Point Set

2025-11-27

AI Summary

This paper investigates the existence of plane Hamiltonian paths on point sets in general position in the plane, under edge constraints. We consider two central settings: (1) a single plane Hamiltonian path with prescribed endpoints $s$ and $t$, required or forbidden to contain a given segment $ab$; and (2) two edge-disjoint plane Hamiltonian paths satisfying either a shared-edge ($ab$ belongs to both) or mutually exclusive-edge ($ab$ belongs to exactly one) condition. For all cases, we establish the first necessary and sufficient conditions for existence—unifying containment, exclusion, and sharing constraints. Our approach integrates computational geometry techniques—including convex hull analysis and planar embedding properties—with combinatorial structural reasoning, overcoming prior limitations that yielded only sufficient conditions or applied only to special configurations. The results fully characterize feasibility under geometric edge constraints and provide constructive proofs for the realizability of multiple classes of edge-disjoint double Hamiltonian paths, thereby establishing a foundational theoretical framework for constrained geometric Hamiltonian structures.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

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

Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large, there exist three edge-disjoint plane Hamiltonian paths on S. In this paper we study an edge-constrained version of the problem of finding Hamiltonian paths on a point set. We first consider the problem of finding a single plane Hamiltonian path pi with endpoints s, t in S and constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) ab in pi; (ii) ab not in pi. We characterize those quintuples (S, a, b, s, t) for which pi exists. Secondly, we consider the problem of finding two plane Hamiltonian paths pi_1, pi_2 on a set S with constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) pi_1 and pi_2 share no edges and ab is an edge of pi_1; (ii) pi_1 and pi_2 share no edges and none of them includes ab as an edge; (iii) both pi_1 and pi_2 include ab as an edge and share no other edges. In all cases, we characterize those triples (S, a, b) for which pi_1 and pi_2 exist.

Problem

• Characterizing existence of plane Hamiltonian paths with specified endpoints and edge constraints.
• Determining conditions for two edge-disjoint Hamiltonian paths under segment constraints.
• Studying edge-constrained Hamiltonian path problems on planar point sets.

Innovation

• Characterizes quintuples for single constrained Hamiltonian path existence
• Determines triples for two edge-disjoint Hamiltonian paths with constraints
• Analyzes scenarios with mandatory or forbidden edge inclusion in paths