Debra L. Boutin

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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)
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(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)