Invariants of almost embeddings of graphs in the plane: results and problems

TL;DR

This paper introduces and studies integer invariants of almost embeddings of graphs in the plane, exploring their properties, realizability, and relations, with implications for understanding graph embeddings and related topological invariants.

Contribution

It defines new invariants for almost embeddings, constructs examples realizing these invariants, and investigates their relations and realizability beyond standard embeddings.

Findings

Defined integer invariants: winding number, Wu numbers.

Constructed almost embeddings with specific invariant values.

Established relations between the invariants and their realizability.

Abstract

A graph drawing in the plane is called an almost embedding if images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce integer invariants of almost embeddings: winding number, cyclic and triodic Wu numbers. We construct almost embeddings realizing some values of these invariants. We prove some relations between the invariants. We study values realizable as invariants of some almost embedding, but not of any embedding. This paper is expository and is accessible to mathematicians not specialized in the area (and to students). However elementary, this paper is motivated by frontline of research.