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A set of vertices S is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of a graph is the size of a smallest determining set. This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to… (More)

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… (More)

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. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G = G k 1 1 2 · · · 2 G km m is the prime factor decomposition of a connected graph then Det(G) = max{Det(G… (More)

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.distinguishing if no nontrivial geometric automorphism preserves the labels. The distinguishing number of G is the minimum r such that G has an… (More)

A geometric automorphism is an automorphism of a geometric graph that preserves crossings and noncrossings of edges. We prove two theorems constraining the action of a geometric automorphism on the boundary of the convex hull of a geometric clique. First, any geometric automorphism that fixes the boundary of the convex hull fixes the entire clique. Second,… (More)

The purpose of this paper is to offer new insight and tools toward the pursuit of the largest chromatic number in the class of thickness-two graphs. At present, the highest chromatic number known for a thickness-two graph is 9, and there is only one known color-critical 1 such graph. We introduce 40 small 9-critical thickness-two graphs, and then use a new… (More)

A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S… (More)

Dedicated to Carsten Thomassen on the occasion of his 60th birthday. Abstract A graph has thickness t if the edges can be decomposed into t and no fewer planar layers. We study one aspect of a generalization of Ringel's famous Earth-Moon problem: what is the largest chromatic number of any thickness-2 graph? In particular, given a graph G we consider the… (More)

A geometric graph G is a simple graph drawn on points in the plane, in general position, with straightline edges. A geometric homo-morphism from G to H is a vertex map that preserves adjacencies and crossings. This work proves some basic properties of geometric homo-morphisms and defines the geochromatic number as the minimum n so that there is a geometric… (More)

A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the label classes. The minimum size of a label class in any such labeling of G is called the cost of 2-distinguishing G and is denoted by ρ(G). The determining number of a graph G, denoted Det(G), is the minimum… (More)