Strong Identification Codes for Graphs

TL;DR

This paper investigates strong identification codes in graphs, focusing on their minimum size and existence conditions for given parameters using probabilistic methods.

Contribution

It introduces the concept of strong identification codes with an index and applies probabilistic techniques to analyze their properties and existence.

Findings

Bounds on the minimum size of strong identification codes

Existence results for graphs with specific identification code indices

Application of probabilistic methods to code existence

Abstract

For any graph~\(G,\) a set of vertices~\({\cal V}\) is said to be dominating if every vertex of~\(G\) contains at least one node of~\(G\) and separating if each vertex~\(v\) contains a unique neighbour~\(u_v \in {\cal V}\) that is adjacent to no other vertex of~\(G.\) If~\({\cal V}\) is both dominating and separating, then~\({\cal V}\) is defined to be an identification code. In this paper, we study strong identification codes with an index~\(r,\) by imposing the constraint that each vertex of~\(G\) contains at least~\(r\) unique neighbours in~\({\cal V}.\) We use the probabilistic method to study both the minimum size of strong identification codes and the existence of graphs that allow an identification code with a given index.