Automorphisms and Distinguishing Numbers of Geometric Cliques

Let G denote a geometric graph. In particular, V (G) is a set of points in general position in R and the edge uv ∈ E(G) is the straight line segment joining the corresponding pair of points. Two edges, say uv and xy, are said to cross if the interiors of the line segments from u to v and x to y have nonempty intersection. A bijection from V (G) to itself is called a geometric automorphism if it preserves adjacency and non-adjacency of vertices, as well as crossing and non-crossing of edges. We let Kn denote a geometric clique (or a geometric complete graph) on n vertices. It is convenient to denote the boundary of the convex hull of Kn by C. We begin by presenting two theorems describing constraints of the action of a geometric automorphism on C.