D. W. Barnette

Bio: D. W. Barnette is an academic researcher from University of California, Davis. The author has contributed to research in topics: Polytope & Line at infinity. The author has an hindex of 2, co-authored 2 publications receiving 15 citations.

Generating closed 2-cell embeddings in the torus and the projective plane

Abstract: If a graphG is embedded in a manifoldM such that all faces are cells bounded by simple closed curves we say that this is a closed 2-cell embedding ofG inM. We show how to generate the 2-cell embeddings in the projective plane from two minimal graphs and the 2-cell embeddings in the torus from six minimal graphs by vertex splitting and face splitting.

Wv paths on the torus

Abstract: We say that a cell complex has theWv property provided any two vertices can be joined by a path that never returns to a facet once it leaves it. The boundaries of convex polytopes have been shown to have theWv property for polytopes of dimensions at most 3. We extend the three-dimensional result to polyhedral maps on the torus by showing that all such maps have theWv property. We also show that theWv property for all polyhedral maps on manifolds of a given genus is equivalent to the property that for all maps on manifolds of that genus each two faces lie in a subcomplex that is a cell.

Planarizing Graphs | A Survey and Annotated Bibliography

Abstract: Given a nite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with on-line algorithms.

Face-width of embedded graphs

Abstract: In their work on graph minors, Robertson and Seymour introduced the face-width (or representativity) as a measure of how densely a graph is embedded on a surface. The face-width of a graph embedded in S is the smallest number k such that S contains a noncontractibl e closed curve that intersects the graph in k points. We survey recent developments in the theory of embeddings of graphs in surfaces that concern this important invariant of embedded graphs.

CHAPTER 2.4 – Polyhedral Manifolds

Abstract: Publisher Summary This chapter reviews polyhedral manifolds or briefly polyhedra that are compact 2-manifolds built up of planar convex or non-convex polygons. This topic is closely related to combinatorial topology, PL-topology, combinatorics, and graph theory. However, the underlying topology is mainly elementary, and the basic ideas and constructions are similar to those of the theory of convex polytopes. The chapter discusses that a map is called regular if its combinatorial automorphism group acts transitively on its flags. A polyhedral embedding of a regular map is called a combinatorially regular polyhedron. Each combinatorially regular polyhedral map on the 2-sphere is combinatorially equivalent to one of the 5 platonic solids and hence possesses even a geometrically regular realization. Also, weaker polyhedral variants besides the combinatorially regular polyhedra are of some interest.