Distinguishing generalized Mycielskian graphs

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, Dist(G). The My-cielskian of a graph G, mu(G), is constructed by adding a shadow vertex u(i) for each vertex vi of G, one additional vertex w, and edges so that N(u(i)) = N-G(v(i)). {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 is not an element of{K-1, K-2} and the number of isolated vertices in mu(t)(G) is at most Dist(G), then Dist (mu(t)(G)) <= Dist(G). This result proves and exceeds a conjecture of Alikhani and Soltani. (C) The author(s).