Properties of leveled spatial graphs

TL;DR

This paper explores the properties of leveled spatial graphs, introduces new invariants to measure their complexity, and analyzes their relation to planarity and other graph invariants.

Contribution

It characterizes graphs with leveled embeddings, introduces the level number and Hamiltonian level number, and relates these to planarity and existing graph invariants.

Findings

All leveled embeddings are free.

Characterization of graphs with low level number.

Determination of level numbers for complete and bipartite graphs.

Abstract

We investigate the property of a spatial graph of having a leveled embedding and characterize the abstract graphs with this property. We show that all leveled embeddings are free and we compare leveled and paneled (also known as flat) embeddings. Two new graph invariants are introduced: the level number, an invariant for graphs that admit a leveled embedding, and the Hamiltonian level number, an invariant for Hamiltonian graphs. These invariants provide a measure on how far a graph is from being planar. We study the relation between the (Hamiltonian) level number and other graph invariants that minimize the decomposition of a graph in planar subgraphs, namely the thickness and the book thickness of a graph. We characterize graphs with low level number and determine both the level number and the Hamiltonian level number of complete graphs and of complete bipartite graphs.