Distinguishing geometric graphs

We begin the study of distinguishing geometric graphs. Let G be a geometric graph. An automorphism of the underlying graph that preserves both crossings and noncrossings is called a geometric automorphism. A labelling, f : V (G) → {1, 2, . . . , r}, is said to be rdistinguishing if no nontrivial geometric automorphism preserves the labels. The distinguishing number of G is the minimum r such that G has an r-distinguishing labelling. We show that when Kn is not the nonconvex K4, it can be 3-distinguished. Furthermore when n ≥ 6, there is a Kn that can be 1-distinguished. For n ≥ 4, K2,n can realize any distinguishing number between 1 and n inclusive. Finally we show that every K3,3 can be 2-distinguished. We also offer several open questions.