Line crossing minimization on metro maps

Authors:
Michael A. Bekos;Michael Kaufmann;Katerina Potika;Antonios Symvonis
Affiliations:
National Technical University of Athens, School of Applied Mathematics & Physical Sciences, Athens, Greece;University of Tübingen, Institute for Informatics, Tübingen, Germany;National Technical University of Athens, School of Applied Mathematics & Physical Sciences, Athens, Greece;National Technical University of Athens, School of Applied Mathematics & Physical Sciences, Athens, Greece
Venue:
GD'07 Proceedings of the 15th international conference on Graph drawing
Year:
2007

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Abstract

We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V,E), so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway line which connects them, whereas the paths illustrate the lines connecting terminal stations. We call this the metro-line crossing minimization problem (MLCM). In contrast to the problem of drawing the underlying graph nicely, MLCM has received fewer attention. It was recently introduced by Benkert et. al in [4]. In this paper, as a first step towards solving MLCM in arbitrary graphs, we study path and tree networks.We examine several variations of the problem for which we develop algorithms for obtaining optimal solutions.