Matchings in Geometric Graphs

Ahmad Biniaz

Home/Archive/arXiv

arXiv · 1610.06457

Matchings in Geometric Graphs

Ahmad Biniaz

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CG

Keep this discovery

BibTeX RIS

Ahmad Biniaz. 2016-10-20. Matchings in Geometric Graphs. https://arxiv.org/abs/1610.06457

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Almost Linear Universal Point Sets for Planar Graphs

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

cs.CG↗

arXiv2026

Some results on Archdeacon's conjecture for rotation systems

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

cs.CG↗

arXiv2026

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

cs.CG↗