  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.
