Augmenting Plane Straight-Line Graphs to Meet Parity Constraints

TL;DR

This paper presents polynomial-time algorithms for augmenting plane geometric graphs to meet specific parity degree constraints, solving open problems for convex position and path graphs.

Contribution

The paper introduces efficient algorithms for augmenting plane graphs to satisfy parity constraints, including a linear-time solution for convex position and an O(n log n) algorithm for paths.

Findings

Polynomial-time decision algorithms for supergraph existence

Linear-time algorithm for convex position graphs

O(n log n) algorithm for plane geometric paths

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).