Stabilizing on the distinguishing number of a graph

TL;DR

This paper investigates the stability of the distinguishing number of graphs under vertex removal, establishing bounds and relationships, and also explores the edge distinguishing stability number.

Contribution

It introduces bounds on the distinguishing stability, relates stability of a graph to that of its subgraphs, and studies the edge distinguishing stability number.

Findings

Established an upper bound for the distinguishing stability: $st_D(G) \\leq |V(G)| - D(G) + 1$.

Proved a relationship between the stability of a graph and its vertex-removed subgraphs: $st_D(G) \\leq st_D(G-v) + 1$.

Explored the edge distinguishing stability number (distinguishing bondage number) of graphs.

Abstract

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing stability, of a graph $G$ is denoted by $st_D(G)$ and is the minimum number of vertices whose removal changes the distinguishing number. We obtain a general upper bound $st_D(G) \leqslant \vert V(G)\vert -D(G)+1$, and a relationships between the distinguishing stabilities of graphs $G$ and $G-v$, i.e., $st_D(G)\leqslant st_D(G-v)+1$, where $v\in V(G)$. Also we study the edge distinguishing stability number (distinguishing bondage number) of $G$.