[Paper Review] Sparse universal graphs for planarity

Louis Esperet, Gwenaël Joret+1 more|arXiv (Cornell University)|Oct 12, 2020

Limits and Structures in Graph Theory4 citations

TL;DR

This paper constructs sparse universal graphs for $n$-vertex planar graphs with $n^{1+o(1)}$ edges, significantly improving the prior $O(n^{3/2})$ bound. It leverages the product structure theorem for planar graphs (as subgraphs of $H \boxtimes P$ with $H$ of treewidth at most 8) and introduces a novel encoding using bulk tree sequences to achieve near-linear edge count, while also constructing an induced-universal graph with $n^{1+o(1)}$ vertices and edges.

ABSTRACT

We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $(1+o(1))n$ vertices and $n^{1 + o(1)}$ edges such that every $n$-vertex planar graph is isomorphic to a subgraph of $G_n$. The best previous bound on the number of edges was $O(n^{3/2})$, proved by Babai, Chung, Erdős, Graham, and Spencer in 1982. We then show that for every integer $n\geq 1$ there is a graph $U_n$ with $n^{1 + o(1)}$ vertices and edges that contains induced copies of every $n$-vertex planar graph. This significantly reduces the number of edges in a recent construction of the authors with Dujmović, Gavoille, and Micek.

Motivation & Objective

• To construct a universal graph for $n$-vertex planar graphs with near-linear edge count, improving upon the longstanding $O(n^{3/2})$ bound.
• To develop an induced-universal graph for $n$-vertex planar graphs with $n^{1+o(1)}$ vertices and edges, achieving both sparsity and induced subgraph containment.
• To generalize the construction to graphs of bounded treewidth and apex-minor-free classes using strong product decompositions.
• To reduce the number of edges in induced-universal graphs while maintaining near-linear size, addressing a key gap in prior constructions.

Proposed method

• Uses the product structure theorem: every planar graph is a subgraph of $H \boxtimes P$ for some $H$ of treewidth at most 8 and a path $P$, as established by Dujmović et al. (2012).
• Employs bulk tree sequences to efficiently encode rows in the product structure, minimizing changes between consecutive rows and enabling compact representation.
• Constructs a universal graph $U_n$ by encoding all $n$-vertex graphs in $\mathcal{Q}_t$ (graphs of the form $H \boxtimes P$ with $H$ of treewidth $t$) using bitstrings and tree sequences.
• Applies a labeling scheme based on transition codes $\mu_y(v)$ and height-bounded trees to control edge count and ensure subgraph containment.
• Uses a tradeoff between transition code length $|\mu_y(v)|$ and tree height to optimize the $n^{1+o(1)}$ edge bound, with optimal $|\mu_y(v)| \in O(\sqrt{\log n \log \log n})$.
• Replaces bulk tree sequences with B-trees (from Gawrychowski and Janczewski, 2023) to slightly improve the edge bound to $n \cdot 2^{O(\sqrt{\log n})}$.

Experimental results

Research questions

• RQ1 Can a universal graph for $n$-vertex planar graphs be constructed with $n^{1+o(1)}$ edges, significantly improving the $O(n^{3/2})$ bound from 1982?
• RQ2 Is it possible to construct an induced-universal graph for $n$-vertex planar graphs with both $n^{1+o(1)}$ vertices and $n^{1+o(1)}$ edges?
• RQ3 Can the construction be generalized to other graph classes, such as apex-minor-free or bounded genus graphs, using the product structure framework?
• RQ4 What is the optimal tradeoff between transition code length and tree height in the encoding scheme to minimize edge count?

Key findings

• A universal graph for $n$-vertex planar graphs exists with $(1+o(1))n$ vertices and at most $n \cdot 2^{O(\sqrt{\log n \cdot \log \log n})}$ edges, improving the prior $O(n^{3/2})$ bound.
• An induced-universal graph for $n$-vertex planar graphs exists with $n^{1+o(1)}$ vertices and $n^{1+o(1)}$ edges, achieving both near-linear size and sparsity.
• The edge count can be further improved to $n \cdot 2^{O(\sqrt{\log n})}$ by replacing bulk tree sequences with B-trees, as shown in recent follow-up work.
• The construction applies to all graphs in $\mathcal{Q}_t$, the class of graphs $H \boxtimes P$ with $H$ of treewidth $t$, yielding a universal graph with $t^2 \cdot n \cdot 2^{O(\sqrt{\log n \cdot \log \log n})}$ edges.
• A labeling scheme for graphs of treewidth at most $t$ yields a graph $H_n$ with $n(\log n)^{O(t^2)}$ vertices and edges that contains all $n$-vertex subgraphs of treewidth $\leq t$ as induced subgraphs.
• The method establishes that the bottleneck in edge count lies in the tradeoff between transition code length and tree height, optimized at $O(\sqrt{\log n \log \log n})$.