Optimal induced universal graphs and adjacency labeling for trees

TL;DR

This paper constructs compact universal graphs for forests and trees, providing optimal adjacency labeling schemes that solve longstanding open problems in graph theory and graph encoding.

Contribution

It introduces the first linear-size universal graph for forests and an optimal adjacency labeling scheme for trees, matching known lower bounds.

Findings

Existence of O(n) node universal graph for forests

Optimal adjacency labeling scheme with log n + O(1) bits

Resolution of a decades-old open problem in graph encoding

Abstract

We show that there exists a graph $G$ with $O(n)$ nodes, where any forest of $n$ nodes is a node-induced subgraph of $G$. Furthermore, for constant arboricity $k$, the result implies the existence of a graph with $O(n^k)$ nodes that contains all $n$-node graphs as node-induced subgraphs, matching a $\Omega(n^k)$ lower bound. The lower bound and previously best upper bounds were presented in Alstrup and Rauhe (FOCS'02). Our upper bounds are obtained through a $\log_2 n +O(1)$ labeling scheme for adjacency queries in forests. We hereby solve an open problem being raised repeatedly over decades, e.g. in Kannan, Naor, Rudich (STOC 1988), Chung (J. of Graph Theory 1990), Fraigniaud and Korman (SODA 2010).