Twins and Vertex- Identification on Graphs

TL;DR

This paper investigates how the presence of twins in graphs affects the optimal values of various vertex-identifying parameters, extending the understanding of identifying colorings beyond twin-free graphs.

Contribution

It introduces the impact of twins on vertex-identifying parameters, providing new insights into identifying coloring and locally identifying coloring in graphs with twins.

Findings

Twins can significantly alter the optimal values of vertex-identifying parameters.

The study extends existing theories to include graphs with twins.

Results show differences in identifying colorings when twins are present.

Abstract

Recently, several vertex identifying notions were introduced (identifying coloring, lid-coloring, ...), these notions were inspired by identifying codes. All of them, as well as original identifying code, are based on separating two vertices according to some conditions on their closed neighborhood. Therefore, twins can not be identified. So most of known results focus on twin-free graph. Here, we show how twins can modify optimal value of vertex-identifying parameters for identifying coloring and locally identifying coloring. \textbf{Keyword :} Identifying coloring, locally identifying coloring, twins, separating.