Optimal labelling schemes for adjacency, comparability, and reachability

TL;DR

This paper develops asymptotically optimal labeling schemes for adjacency, reachability, and comparability in large hereditary graph classes, achieving minimal label sizes of roughly n/4 bits per vertex, which are proven to be optimal.

Contribution

It introduces the first asymptotically optimal labeling schemes for hereditary graph classes with exponential size, including perfect and comparability graphs, with minimal label sizes.

Findings

Labels of n/4+o(n) bits per vertex for adjacency in perfect and comparability graphs.

Existence of reachability and comparability labeling schemes with similar label sizes.

Results are proven to be optimal up to lower order terms.

Abstract

We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing $2^{\Omega(n^2)}$ $n$-vertex graphs as $n\to \infty$. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an adjacency labelling scheme with labels of $n/4+o(n)$ bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of $n/4+o(n)$ bits per vertex and comparability labelling scheme for posets with labels of $n/4+o(n)$ bits per element. All these results are best possible, up to the lower order term.