Distinguishing Generalized Mycielskian Graphs

Abstract

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.