Symmetry breaking in planar and maximal outerplanar graphs

TL;DR

This paper investigates the symmetry-breaking properties of maximal outerplanar graphs, Halin graphs, and Mycielskian graphs, showing that most can be distinguished with at most two labels, and computes specific distinguishing parameters.

Contribution

It proves that all maximal outerplanar graphs except K3 have a distinguishing number and index at most two, and calculates these parameters for Halin and Mycielskian graphs.

Findings

Maximal outerplanar graphs (except K3) are distinguishable with two labels.

The distinguishing number and index of Halin and Mycielskian graphs are explicitly computed.

Most studied graphs can be uniquely identified by simple labelings, reducing symmetry.

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has a vertex (edge) labeling with $d$ labels that is preserved only by a trivial automorphism. In this paper we consider the maximal outerplanar graphs (MOP graphs) and show that MOP graphs, except $K_3$, can be distinguished by at most two vertex (edge) labels. We also compute the distinguishing number and the distinguishing index of Halin and Mycielskian graphs.