Invariants of almost embeddings of graphs in the plane

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.

Figures (29)

• Figure 1.1: Непланарные графы $K_5$ и $K_{3,3}$
• Figure 1.2: Вложение, почти вложение и отображение (изображение), которое не является почти вложением
• Figure 1.3: Вложение и почти вложение графа $K_5$ без ребра
• Figure 2.1: $w(ABC) =\dfrac{1}{2\pi} \left( \angle AOB + \angle BOC + \angle COA \right) = +1$
• Figure 2.2: Числа оборотов равны $0,~+1,~-1,~+2$

Theorems & Definitions (25)

• proof : Построение
• proof
• proof : Построение примера \ref{['e:k23']}
• proof : Более явное построение примера \ref{['e:k23']}
• proof : Доказательство утверждения \ref{['p:rel1']}
• proof : Доказательство
• proof : Построение примера \ref{['e:maps']} для $n_2 = n_3 = n_4 = 0$
• proof : Построение примера \ref{['e:maps']}
• proof : Более явное построение примера \ref{['e:maps']}
• proof : Доказательство