Identifying codes in vertex-transitive graphs and strongly regular graphs

TL;DR

This paper investigates the fractional and integer identifying codes in vertex-transitive and strongly regular graphs, providing bounds, exact solutions for specific classes, and constructing infinite families with particular code sizes.

Contribution

It introduces new bounds for the ratio of integer to fractional identifying codes and constructs infinite families of graphs with specific code sizes.

Findings

The ratio between optimal integer and fractional solutions is bounded by 1 and 2 ln(|V|)+1.

Exact fractional solutions are computed for vertex-transitive graphs.

Infinite families of graphs with identifying codes of order |V|^a for a in {1/4,1/3,2/5} are constructed.

Abstract

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.