On Vertex and Empty-ply Proximity Drawings

Angelini, Patrizio and Chaplick, Steven and De Luca, Felice and Fiala, Jiří and Hančl Jr., Jaroslav and Heinsohn, Niklas and Kaufmann, Michael and Kobourov, Stephen G. and Kratochvíl, Jan and Valtr, Pavel (2017) On Vertex and Empty-ply Proximity Drawings. In: Graph Drawing and Network Visualization, GD 2017, September 25-27 , pp. 24-37(Official URL: https://doi.org/10.1007/978-3-319-73915-1_3).

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We initiate the study of the vertex-ply of straight-line drawings, as a relaxation of the recently introduced ply number. Consider the disks centered at each vertex with radius equal to half the length of the longest edge incident to the vertex. The vertex-ply of a drawing is determined by the vertex covered by the maximum number of disks. The main motivation for considering this relaxation is to relate the concept of ply to proximity drawings. In fact, if we interpret the set of disks as proximity regions, a drawing with vertex-ply number 1 can be seen as a weak proximity drawing, which we call empty-ply drawing. We show non-trivial relationships between the ply number and the vertex-ply number. Then, we focus on empty-ply drawings, proving some properties and studying what classes of graphs admit such drawings. Finally, we prove a lower bound on the ply and the vertex-ply of planar drawings.

Item Type: Conference Paper
Classifications: G Algorithms and Complexity > G.560 Geometry
P Styles > P.720 Straight-line
URI: http://gdea.informatik.uni-koeln.de/id/eprint/1590

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