Trees with distinguishing number two

TL;DR

This paper characterizes all trees with radius at most three that have a distinguishing number of two, and provides a necessary condition for such trees with larger radius, advancing understanding of graph symmetry breaking.

Contribution

It offers a complete characterization of trees with radius at most three and distinguishing number two, and introduces a necessary condition for trees with larger radius.

Findings

All trees with radius ≤ 3 and distinguishing number 2 are characterized.

A necessary condition for trees with radius > 3 and distinguishing number 2 is established.

The results enhance understanding of symmetry-breaking in trees based on radius and labeling.

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. In this paper we characterize all trees with radius at most three and distinguishing number two. Also we present a necessary condition for trees with distinguishing number two and radius more than three.