Matchings in Geometric Graphs

TL;DR

This thesis explores various matching problems in geometric graphs, including maximum, bottleneck, plane, and strong matchings, providing bounds and algorithms across different geometric graph families.

Contribution

It introduces new bounds and algorithms for multiple matching problems in diverse geometric graphs, advancing understanding of their combinatorial properties.

Findings

Established bounds for maximum matchings in geometric graphs

Developed algorithms for non-crossing and strong matchings

Identified tight bounds and areas for further refinement

Abstract

A geometric graph is a graph whose vertex set is a set of points in the plane and whose edge set contains straight-line segments. A matching in a graph is a subset of edges of the graph with no shared vertices. A matching is called perfect if it matches all the vertices of the underling graph. A geometric matching is a matching in a geometric graph. In this thesis, we study matching problems in various geometric graphs. Among the family of geometric graphs we look at complete graphs, complete bipartite graphs, complete multipartite graphs, Delaunay graphs, Gabriel graphs, and $\Theta$-graphs. The classical matching problem is to find a matching of maximum size in a given graph. We study this problem as well as some of its variants on geometric graphs. The bottleneck matching problem is to find a maximum matching that minimizes the length of the longest edge. The plane matching problem is to find a maximum matching so that the edges in the matching are pairwise non-crossing. A geometric matching is strong with respect to a given shape $S$ if we can assign to each edge in the matching a scaled version of $S$ such that the shapes representing the edges are pairwise disjoint. The strong matching problem is to find a maximum strong matching with respect to a given shape. The matching packing problem is to pack as many edge-disjoint perfect matchings as possible into a geometric graph. We study these problems and establish lower and upper bounds on the size of different kinds of matchings in various geometric graphs. We also present algorithms for computing such matchings. Some of the presented bounds are tight, while the others need to be sharpened.