DeepAI

# Augmenting Geometric Graphs with Matchings

We study noncrossing geometric graphs and their disjoint compatible geometric matchings. Given a cycle (a polygon) P we want to draw a set of pairwise disjoint straight-line edges with endpoints on the vertices of P such that these new edges neither cross nor contain any edge of the polygon. We prove NP-completeness of deciding whether there is such a perfect matching. For any n-vertex polygon, with n > 3, we show that such a matching with less than n/7 edges is not maximal, that is, it can be extended by another compatible matching edge. We also construct polygons with maximal compatible matchings with n/7 edges, demonstrating the tightness of this bound. Tight bounds on the size of a minimal maximal compatible matching are also obtained for the families of d-regular geometric graphs for each d in 0,1,2. Finally we consider a related problem. We prove that it is NP-complete to decide whether a noncrossing geometric graph G admits a set of compatible noncrossing edges such that G together with these edges has minimum degree five.

• 16 publications
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• 7 publications
03/17/2019

### Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Given a planar straight-line graph G=(V,E) in R^2, a circumscribing poly...
11/17/2022

### Crossing and intersecting families of geometric graphs on point sets

Let S be a set of n points in the plane in general position. Two line se...
07/23/2020

### Plane augmentation of plane graphs to meet parity constraints

A plane topological graph G=(V,E) is a graph drawn in the plane whose ve...
06/28/2017

### On Compatible Triangulations with a Minimum Number of Steiner Points

Two vertex-labelled polygons are compatible if they have the same clockw...
08/20/2022

### Minimizing Visible Edges in Polyhedra

We prove that, given a polyhedron 𝒫 in ℝ^3, every point in ℝ^3 that does...
08/14/2019

### Minimal Representations of Order Types by Geometric Graphs

In order to have a compact visualization of the order type of a given po...
06/29/2018

### (k,p)-Planarity: A Generalization of Hybrid Planarity

A graph G is (k,p)-planar if its vertices can be partitioned into cluste...