Endomorphism Breaking in Graphs

Wilfried Imrich,Florian Lehner,Rafał Kalinowski,Monika Pilśniak

doi:10.37236/3073

Abstract

We introduce the endomorphism distinguishing number $D_e(G)$ of a graph $G$ as the least cardinal $d$ such that $G$ has a vertex coloring with $d$ colors that is only preserved by the trivial endomorphism. This generalizes the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined for automorphisms instead of endomorphisms.As the number of endomorphisms can vastly exceed the number of automorphisms, the new concept opens challenging problems, several of which are presented here. In particular, we investigate relationships between $D_e(G)$ and the endomorphism motion of a graph $G$, that is, the least possible number of vertices moved by a nontrivial endomorphism of $G$. Moreover, we extend numerous results about the distinguishing number of finite and infinite graphs to the endomorphism distinguishing number.

Highlights

Albertson and Collins [1] introduced the distinguishing number D(G) of a graph G as the least cardinal d such that G has a vertex labeling with d labels that is only preserved by the trivial automorphism.This concept has spawned numerous papers, mostly on finite graphs
As the number of endomorphisms can vastly exceed the number of automorphisms, the new concept opens challenging problems, several of which are presented here
Countable infinite graphs have been investigated with respect to the distinguishing number; see [8], [14], [15], and [16]

Summary

Introduction

Albertson and Collins [1] introduced the distinguishing number D(G) of a graph G as the least cardinal d such that G has a vertex labeling with d labels that is only preserved by the trivial automorphism. This concept has spawned numerous papers, mostly on finite graphs. Countable infinite graphs have been investigated with respect to the distinguishing number; see [8], [14], [15], and [16]. The aim of this paper is the presentation of fundamental results for the endomorphism distinguishing number, and of open problems. We extend the Motion Lemma of Russell and Sundaram [13] to endomorphisms, present endomorphism motion conjectures that generalize the Infinite Motion Conjecture of Tom Tucker [15] and the Motion Conjecture of [4], prove the validity of special cases, and support the conjectures by examples

Definitions and Basic Results

The Endomorphism Motion Lemma

Infinite graphs

Examples and outlook