Invariants of almost embeddings of graphs in the plane

TL;DR

This paper investigates invariants of almost embeddings of graphs in the plane, exploring their relations, connections to homology, and constructing examples, while making topological concepts accessible to non-specialists.

Contribution

It establishes relations between invariants of almost embeddings, connects these to homology of the deleted product, and constructs realizations, advancing understanding in topological graph embeddings.

Findings

Relations between invariants are established.

Connections to homology of the deleted product are demonstrated.

Examples of almost embeddings realizing specific invariants are constructed.

Abstract

A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in combinatorial geometry, in topological combinatorics, and in studies of embeddings. We prove some relations between the invariants. We demonstrate the connection of some of these relations to homology of the deleted product of a graph. We construct almost embeddings realizing some values of these invariants. We present some ideas of algebraic and geometric topology in a language accessible to non-topologists (in particular, to students). All the necessary definitions are recalled. However elementary, this paper is motivated by frontline of research; there are some conjectures and open problems.