Introducing Spacial Graphs (updated version)

Luccio, Fabrizio and Pagli, Linda (1995) Introducing Spacial Graphs (updated version). Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.

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Abstract

We introduce spatial graphs through a natural extension of the concept of planarity in three dimensions. For a graph G=(V,E), a face is a simple cycle of edges, and a complete set of faces is such that each edge belonging to a cycle in E is on at least one face in the set. G is spatial if admits a three-dimensional representation R with a complete set of faces consisting of simple surfaces, such that no two faces intersect except along the common edge. In particular G is called k-spatial if R includes k cells (spatial regions); and is called plane-face spatial if all the faces in R are plane. We prove that all graphs are 1-spatial, but not all of them are (k>1)-spatial or plane-face spatial. In particular, $K_n$ is $(n^2-3n)/2$-spatial and plane-face spatial. We derive some basic properties of spatial graphs as natural three-dimensional extensions of the properties of planar graphs.

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