Outerspatial 2-complexes: Extending the class of outerplanar graphs to three dimensions

TL;DR

This paper introduces outerspatial 2-complexes, a three-dimensional extension of outerplanar graphs, characterizing their structure through forbidden subcomplexes and space minors, with applications to constrained graph embeddings.

Contribution

It defines outerspatial 2-complexes, proves their characterization via forbidden minors, and applies this to nested plane embeddings of graphs.

Findings

Outerspatial 2-complexes are characterized by absence of certain subcomplexes.

A locally 2-connected 2-complex is outerspatial iff it avoids specific minors.

Applications include constrained plane embeddings of graphs.

Abstract

We introduce the class of outerspatial 2-complexes as the natural generalisation of the class of outerplanar graphs to three dimensions. Answering a question of O-joung Kwon, we prove that a locally 2-connected 2-complex is outerspatial if and only if it does not contain a surface of positive genus as a subcomplex and does not have a space minor that is a generalised cone over $K_4$ or $K_{2,3}$. This is applied to nested plane embeddings of graphs; that is, plane embeddings constrained by conditions placed on a set of cycles of the graph.