Lexicographic identifying codes

TL;DR

This paper presents algorithms for finding minimum identifying codes in twin-free graphs efficiently, with different complexities for general and sparse graphs, and proves their correctness and optimality.

Contribution

It introduces new algorithms for identifying codes in twin-free graphs, including an efficient method for sparse graphs and a proof of optimality under proper vertex sorting.

Findings

Algorithm finds identifying codes in O(n^3) time for twin-free graphs.

An improved algorithm operates in O(n^2d log n) for sparse graphs.

Algorithms can return minimum cardinality codes if vertices are sorted correctly.

Abstract

An identifying code in a graph is a set of vertices which intersects all the symmetric differences between pairs of neighbourhoods of vertices. Not all graphs have identifying codes; those that do are referred to as twin-free. In this paper, we design an algorithm that finds an identifying code in a twin-free graph on n vertices in O(n^3) binary operations, and returns a failure if the graph is not twin-free. We also determine an alternative for sparse graphs with a running time of O(n^2d log n) binary operations, where d is the maximum degree. We also prove that these algorithms can return any identifying code with minimum cardinality, provided the vertices are correctly sorted.