Extremal graphs for the identifying code problem

TL;DR

This paper classifies all finite and infinite graphs where the minimum identifying code is nearly the entire vertex set, providing new bounds based on graph size and degree.

Contribution

It disproves previous conjectures and fully classifies graphs with minimal identifying codes almost as large as the entire vertex set.

Findings

Classified all finite graphs with minimal identifying code size of n-1.

Classified all infinite graphs requiring all vertices in any identifying code.

Provided new upper bounds based on number of vertices and maximum degree.

Abstract

An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from all other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand that if a graph on n vertices with at least one edge admits an identifying code, then a minimum identifying code has size at most n-1. Some classes of graphs whose smallest identifying code is of size n-1 were already known, and few conjectures were formulated to classify all these graphs. In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We also classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided.