Faces in rectilinear drawings of complete graphs
Martin Balko, Anna Br\"otzner, Fabian Klute, Josef Tkadlec

TL;DR
This paper explores extremal face configurations in convex rectilinear drawings of complete graphs, establishing conditions for convex polygons as faces and characterizing regular drawings with convex 5-gon faces.
Contribution
It introduces the first study of face extremal problems in convex rectilinear drawings of complete graphs and characterizes when regular drawings contain convex 5-gon faces.
Findings
If no common interior point of three edges exists, a convex 5-gon face always exists.
There are convex rectilinear drawings without any convex k-gon face for k ≥ 6.
Characterization of regular drawings containing convex 5-gon faces.
Abstract
We initiate the study of extremal problems about faces in convex rectilinear drawings of~, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex -gon with . A convex rectilinear drawing of is \emph{regular} if its vertices correspond to vertices of a regular convex -gon. We characterize positive integers for which regular drawings of contain a face forming a convex 5-gon. To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Point processes and geometric inequalities · Digital Image Processing Techniques
