Determining Number and Cost of Generalized Mycielskian Graphs

A set S of vertices is a determining set for a graph G if every auto-morphism of G is uniquely determined by its action on S and Det(G) is the size of smallest determining set of G. 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 and Dist(G) is the smallest d for which G is d-distinguishable. If Dist(G) = 2, the cost of 2-distinguishing, rho(G), is the size of a smallest color class over all 2-distinguishing colorings of G. This paper examines the determining number and, when relevant, the cost of 2-distinguishing for Mycielskians mu(G) and generalized Mycielskians mu(t)(G) of simple graphs with no isolated vertices. In particular, if G not equal K-2 is twin-free with no isolated vertices, then Det(mu(t)(G)) = Det(G). If in addition Det(G) >= 2 and t >= Det(G) - 1, then Dist(mu(t)(G)) = 2 and rho(mu(t)(G)) = Det(G). For G with twins, we develop a framework using quotient graphs to find Det(mu(G)) and Det(mu(t)(G)) in terms of Det(G).