Functionality of Random Graphs

TL;DR

This paper investigates the functionality parameter of random graphs, establishing bounds that relate to classical graph parameters, and provides a comprehensive understanding of its behavior across different probabilities.

Contribution

It introduces bounds on the functionality of Erdős–Rényi random graphs, connecting it to well-known graph parameters like degeneracy and twin-width.

Findings

Functionality of G(n,p) is determined up to a constant factor for all p.

Establishes relationships between functionality and classical graph parameters.

Provides bounds that enhance understanding of graph structure in random models.

Abstract

The functionality of a graph $G$ is the minimum number $k$ such that in every induced subgraph of $G$ there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most $k$ other vertices in the subgraph. The functionality parameter was introduced in the context of adjacency labeling schemes, and it generalises a number of classical and recent graph parameters including degeneracy, twin-width, and symmetric difference. We establish the functionality of a random graph $G(n,p)$ up to a constant factor for every value of $p$.