Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets

TL;DR

This paper classifies all finite and infinite digraphs where the entire vertex set is the only identifying code, and relates these findings to extremal cases of Bondy's theorem on set systems.

Contribution

It provides a complete classification of digraphs with trivial identifying codes and connects this to extremal cases of Bondy's theorem, advancing understanding of graph identification and set system extremities.

Findings

Classified all finite digraphs with only the entire vertex set as an identifying code.

Classified all infinite oriented graphs with the same property.

Linked these classifications to extremal cases of Bondy's theorem.

Abstract

An identifying code of a (di)graph $G$ is a dominating subset $C$ of the vertices of $G$ such that all distinct vertices of $G$ have distinct (in)neighbourhoods within $C$. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A. Bondy on set systems we classify the extremal cases for this theorem.