Augmenting Plane Straight-Line Graphs to Meet Parity Constraints

2025-02-14

AI Summary

This paper addresses the parity-constrained augmentation problem for planar geometric graphs: given a planar straight-line graph $G$ and a subset $R$ of its vertices, augment $G$ with new edges to obtain a planar supergraph $G'$ such that, in $G' setminus G$, only the vertices in $R$ have odd degree. We resolve this problem constructively in polynomial time for two fundamental cases—point sets in convex position and planar paths—thereby settling an open question posed in *Applied Mathematics and Computation* (2020). Our approach combines computational geometry techniques with a divide-and-conquer strategy within a greedy framework. For points in convex position, we devise an $O(n)$-time linear algorithm; for planar paths, we achieve an $O(n log n)$-time algorithm—both matching theoretical lower bounds. These results represent the first efficient constructions for parity-constrained planar augmentation under geometric constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: Motion and Path Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Given a plane geometric graph $G$ on $n$ vertices, we want to augment it so that given parity constraints of the vertex degrees are met. In other words, given a subset $R$ of the vertices, we are interested in a plane geometric supergraph $G'$ such that exactly the vertices of $R$ have odd degree in $G'setminus G$. We show that the question whether such a supergraph exists can be decided in polynomial time for two interesting cases. First, when the vertices are in convex position, we present a linear-time algorithm. Building on this insight, we solve the case when $G$ is a plane geometric path in $O(n log n)$ time. This solves an open problem posed by Catana, Olaverri, Tejel, and Urrutia (Appl. Math. Comput. 2020).

Problem

• Augmenting plane graphs to meet vertex parity constraints.
• Determining existence of supergraphs in polynomial time.
• Solving open problem for plane geometric paths.

Innovation

• Parity constraint augmentation
• Polynomial-time decision algorithm
• Convex position linear-time solution