Topological Drawings of Complete Bipartite Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. 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 \(K_{2,2}\) and \(K_{3,2}\), and investigate the constraints they must satisfy. We prove in particular that for complete bipartite graphs of the form \(K_{2,n}\) and \(K_{3,n}\), such an encoding corresponds to a drawing if and only if it obeys consistency conditions on triples and quadruples. In the general case of \(K_{k,n}\) with \(k\ge 2\), we completely characterize and enumerate drawings in which the order of the edges around each vertex is the same for vertices on the same side of the bipartition.

Notes

Acknowledgments

This work was started at the Workshop on Order Types, Rotation Systems, and Good Drawings in Strobl 2015. We thank the organizers and participants for fruitful discussions, in particular Pedro Ramos who suggested to look at complete bipartite graphs. We thank the anonymous referees for their insightful remarks.

S. Felsner also acknowledges support from DFG grant Fe 340/11-1.

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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Université Libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.Technische Universität (TU) BerlinBerlinGermany

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