Different Types of Isomorphisms of Drawings of Complete Multipartite Graphs

TL;DR

This paper investigates the relationships between various characteristics of simple drawings of complete multipartite graphs, establishing which features determine others and defining meaningful isomorphisms for these graph drawings.

Contribution

It characterizes how crossing orders determine other drawing features and identifies the conditions under which certain isomorphisms are meaningful for complete multipartite graphs.

Findings

Edge crossing orders determine all other characteristics.

Partition classes of size at least three ensure certain determinacies.

Most implications from complete graphs do not hold for multipartite graphs.

Abstract

Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge intersects itself. We analyze several characteristics of simple drawings of complete multipartite graphs: which pairs of edges cross, in which order they cross, and the cyclic order around vertices and crossings, respectively. We consider all possible combinations of how two drawings can share some characteristics and determine which other characteristics they imply and which they do not imply. Our main results are that for simple drawings of complete multipartite graphs, the orders in which edges cross determine all other considered characteristics. Further, if all partition classes have at least three vertices, then the pairs of edges that cross determine the rotation system and the rotation around the crossings determine the extended rotation system. We also show that most other implications -- including the ones that hold for complete graphs -- do not hold for complete multipartite graphs. Using this analysis, we establish which types of isomorphisms are meaningful for simple drawings of complete multipartite graphs.