Symmetry Parameters for Mycielskian Graphs

TL;DR

This paper investigates how symmetry parameters such as the determining number, distinguishing number, and cost of distinguishing behave in Mycielskian and generalized Mycielskian graphs, linking these parameters to those of the original graph.

Contribution

It provides new insights into the symmetry parameters of Mycielskian graphs and their generalizations, relating these parameters to the original graph's properties.

Findings

Symmetry parameters are characterized in terms of the original graph.

Results connect the symmetry parameters of $u(G)$ and $u_t(G)$ to those of $G$.

The study advances understanding of symmetry in graph constructions.

Abstract

The Mycielskian construction, denoted $\mu(G)$, takes a finite simple graph $G$ to a larger graph with of the same clique number but larger chromatic number. The generalized Mycielskian construction, denoted $\mu_t(G)$, takes $G$ to a larger graph with the same chromatic number but with larger odd girth. In this chapter we look at symmetry parameters of $\mu(G)$ and $\mu_t(G)$ in terms of the same parameters of $G$. These symmetry parameters include determining number, distinguishing number, and cost of distinguishing.