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Automorphisms and Distinguishing Numbers of Geometric Cliques

Summary: Automorphisms and Distinguishing Numbers of
Geometric Cliques
Michael O. Albertson1
Debra L. Boutin2
Let G denote a geometric graph. In particular, V (G) is a set of points in general position in R2
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.
Theorem 1. Any geometric automorphism that fixes each vertex on the boundary of the convex hull of Kn
fixes every vertex of the graph.
We prove this theorem by assuming there is an automorphism f that fixes the vertices of C, but f(x) = y
for two distinct vertices interior to C. If we repeatedly apply this automorphism we must get fr
(x) = x for
some integer r. A contradiction is realized when we see that this implies a pair of uncrossed edges had to
be crossed in the process.


Source: Albertson, Michael O. - Department of Mathematics and Statistics, Smith College


Collections: Mathematics