Topological Drawings of Complete Bipartite Graphs
Jean Cardinal, Stefan Felsner

TL;DR
This paper studies a specific class of topological drawings of complete bipartite graphs, providing combinatorial encodings, polynomial-time algorithms for realizability, and characterizations of certain symmetric and straight-line drawings.
Contribution
It introduces combinatorial encodings for these drawings, characterizes their existence via local conditions, and develops polynomial-time algorithms for realizability and enumeration.
Findings
Existence of drawings characterized by simple local conditions.
Polynomial-time algorithms for realizability of topological drawings.
Complete enumeration of symmetric bipartite graph drawings.
Abstract
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been studied extensively in the context of crossing number problems. We consider a natural class of simple topological drawings of complete bipartite graphs, in which we require that one side of the vertex set bipartition lies on the outer boundary of the drawing. We investigate the combinatorics of such drawings. For this purpose, we define combinatorial encodings of the drawings by enumerating the distinct drawings of subgraphs isomorphic to and , and investigate the constraints they must satisfy. We prove that a drawing of exists if and only if some simple local conditions are satisfied by the encodings. This directly…
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