Universal Geometric Graphs

TL;DR

This paper constructs minimal geometric graphs that are universal for specific classes of planar graphs, like forests and caterpillars, with tight bounds on edges, advancing understanding of geometric graph universality.

Contribution

It introduces universal geometric graphs with optimal edge bounds for forests and caterpillars, and establishes near-quadratic bounds for outerplanar graphs.

Findings

Existence of an $O(n \, ext{log} \, n)$ edge universal geometric graph for $n$-vertex forests.

Universal convex geometric graphs for outerplanar graphs require near-quadratic edges.

Constructed an $O(n \, ext{log} \, n)$ edge universal convex geometric graph for caterpillars.

Abstract

We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class $\mathcal H$ of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an $n$-vertex graph with $O(n \log n)$ edges that contains every $n$-vertex forest as a subgraph. Our $O(n \log n)$ bound on the number of edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that, for every positive integer $h$, every $n$-vertex convex geometric graph that is universal for $n$-vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega_h(n^{2-1/h})$; this almost matches the trivial $O(n^2)$ upper bound given by the $n$-vertex complete convex geometric graph. Finally, we prove that there exists an $n$-vertex convex geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex caterpillars.