A graph G is said to be d-distinguishable if there is a d-coloring of G which no non-trivial automorphism preserves. That is, 9c : G ! f1; : : : ; dg; 8f 2 Aut(G)nfidg;9v;c(v) 6= c(f(v)): It was conjectured that if jGj ? jAut(G)j and the Aut(G) action on G has no singleton orbits, then G is 2-distinguishable. We give an example where this fails. We partially repair the conjecture by showing that when "enough motion occurs," the distinguishing number does indeed decay. Specifically, defining m(G) = min f2Aut(G) f6=id j fv 2 G : f(v) 6= vg j; we show that when m(G) ? 2log 2 jAut(G)j, G is indeed 2-distinguishable. In general, we show that if m(G)lnd ? 2ln jAut(G)j then G is d-distinguishable. There has been considerable interest in the computational complexity of the d- distinguishability problem. Specifically, there has been much musing on the computational complexity of the language f(G;d) : G is d-distinguishableg : We show that this language lies in AM ae S P 2 " P P 2 ....

Given a graph G, a labeling c: V (G) →{1, 2,...,d} is said to be d-distinguishing if the only element in Aut(G) that preserves the labels is the identity. The distinguishing number of G, denoted by D(G), is the minimum d such that G has a d-distinguishing labeling. If G H denotes the Cartesian product of G and H, let G 2 = G G and G r = G G r−1. A graph G is said to be prime with respect to the Cartesian product if whenever G ∼ = G1 G2, then either G1 or G2 is a singleton vertex. This paper proves that if G is a connected, prime graph, then D(G r)=2 whenever r ≥ 4.

Let G be a graph. A vertex labeling of G is distinguishing if the only label-preserving automorphism of G is the identity map. The distinguishing number of G, D(G), isthe minimum number of labels needed so that G has a distinguishing labeling. In this paper, we present O(n log n)-time algorithms that compute the distinguishing numbers of trees and forests. Unlike most of the previous work in this area, our algorithm relies on the combinatorial properties of trees rather than their automorphism groups to compute for their distinguishing numbers. 1

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted DG(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product G ≀Y H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups Sm × Sn on [m] × [n]. 1

This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels. A graph G is said to be d-distinguishable if there is a labeling of the vertex set using 1,..., d so that no nontrivial automorphism of G preserves the labels. A set of vertices S ⊆ V (G) is a determining set for G if every automorphism of G is uniquely determined by its action on S. We prove that a graph is d-distinguishable if and only if it has a determining set that can be (d − 1)-distinguished. We use this to prove that every Kneser graph Kn:k with n ≥ 6 and k ≥ 2 is 2-distinguishable. 1

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 K n 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, K 2,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. 1

A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph.

The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label G so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups. 1

The distinguishing number of a graph G is the smallest positive integer r such that G has a labeling of its vertices with r labels for which there is no non-trivial automorphism of G preserving these labels. In early work, Michael Albertson and Karen Collins computed the distinguishing number for various finite graphs, and more recently Wilfried Imrich, Sandi Klavzar and Vladimir Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed graphs, and posets. We show that this number is in most cases two or infinite, and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures.

Let G denote a geometric graph. In particular, V (G) is a set of points in general position in R 2 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. 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 f r (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. Theorem 2. If the boundary of the convex hull of Kn contains at least four vertices, then every geometric