  A graph $G$ is $d$-distinguishable if there is a coloring of the vertices
with $d$ colors so that only the trivial automorphism preserves the color
classes. The smallest such $d$ is the distinguishing number,
$\operatorname{Dist}(G)$. The Mycielskian $\mu(G)$ of a graph $G$ is
constructed by adding a shadow vertex $u_i$ for each vertex $v_i$ of $G$ and
one additional vertex $w$ and adding edges so that
$N(u_i)~=~N_G(v_i)~\cup~\{w\}$. The generalized Mycielskian $\mu_t(G)$ is a
Mycielskian graph with $t$ layers of shadow vertices, each with edges to layers
above and below. This paper examines the distinguishing number of the
traditional and generalized Mycielskian graphs. Notably, if $G~\neq ~K_1,~K_2$
and the number of isolated vertices in $\mu_t(G)$ is at most
$\operatorname{Dist}(G)$, then $\operatorname{Dist}(\mu_t(G)) \le
\operatorname{Dist}(G)$. This result proves and exceeds a conjecture of
Alikhani and Soltani.
