Augmenting Plane Straight-Line Graphs to Meet Parity Constraints

TL;DR

This work studies augmenting plane geometric graphs by adding straight-line edges so that prescribed vertex-degree parities are achieved, with the constraint that $|R|$ is even. It ties parity satisfaction to finding compatible geometric spanning structures in visibility graphs, using hugging cycles and tight hulls to enable efficient testing via a weak-dual dynamic program. The authors provide a linear-time algorithm for convex-position graphs and an $O(|V|\log|V|)$-time algorithm for plane geometric paths, solving an open problem of Catana et al. The results offer structural insight and practical algorithms for parity-constrained graph augmentation in planar geometric settings.

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).

Figures (5)

• Figure 1: (a) A plane geometric graph $G$ (in black) and its visibility graph $\mathrm{Vis}(G)$ (in gray). (b) A solution set for $G$ and the four unhappy vertices (red squares).
• Figure 6: An example in which $\mathrm{Vis}(G)$ is not connected and an even set of two unhappy vertices (square) that does not admit a solution.
• Figure 7: Illustration for the proof of \ref{['lem:Sufficient1']}. (a) A plane geometric graph $G$ (in black) and a plane spanning tree $T$ of $\mathrm{Vis}(G)$ (in gray). (b) A set $R$ and paths (in dashed red) between (an arbitrary partition of $R$ into) pairs of $R$ in $T$. (c) The symmetric difference of the paths yields a solution set (in red).
• Figure 8: Example of a path and its pockets. While pocket $P$ (with blue interior) is pseudo-convex, pocket $P'$ (with red interior) is not.
• Figure 9: Example of a pseudo-convex path and the tight hull of a pocket (with blue interior), its subpockets (with darker blue interior), and the tight hull of $G$ forming a hugging cycle (in gray).

Theorems & Definitions (11)

• Theorem 1
• Corollary 2
• Lemma 2
• Lemma 2
• Lemma 2
• Theorem 3
• Lemma 4
• Lemma 5: Theorem 3 and Remark 2 in GARCIA2014
• Theorem 6
• Lemma 7