Sewing Ribbons on Graphs in Space

An open ribbon is a square with one side called the seam. A closed ribbon is a cylinder with one boundary component called the seam. We sew an open (resp. closed) ribbon onto a graph by identifying the seam with an open (resp. closed) walk in the graph. A ribbon complex is a graph with a nite number of ribbons sewn on. We investigate when a ribbon complex embeds in 3-dimensional Euclidean space. We give several characterizations of such spatial complexes which lead to algorithms. We examine special cases where: 1) each edge of the graph is incident with at most three ribbons, and 2) every ribbon is closed together with a connectivity condition.