1. Area-Thickness Trade-Offs for Straight-Line Drawings of Planar Graphs.
- Author
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DI GIACOMO, EMILIO, DIDIMO, WALTER, LIOTTA, GIUSEPPE, and MONTECCHIANI, FABRIZIO
- Subjects
PLANAR graphs ,DRAWING ,GEOMETRIC vertices ,EDGES (Geometry) ,MATHEMATICAL mappings - Abstract
We study the problem of computing drawings of planar graphs in sub-quadratic area, by allowing edge crossings. We first prove that sub-quadratic area cannot be achieved if only a constant number of crossings per edge is allowed. More precisely, we show that the same area lower bounds as in the crossing-free case hold for straight-line and poly-line drawings of planar graphs and seriesparallel graphs. Motivated by this result, we study straight-line drawings of planar graphs where the number of crossings per edge is not bounded by a constant. In this case, we prove that every planar graph admits a straight-line drawing with sub-quadratic area and sub-linear thickness (the thickness of a drawing is the minimum number of colors that can be assigned to the edges so that each color class induces a planar drawing). We also prove that every partial 2-tree (and hence every series-parallel graph) admits a linear-area straight-line drawing with thickness at most 10. It is worth remarking that a drawing with thickness h - 1 is h-quasi planar, i.e. it does not contain h-mutually crossing edges. The main ingredient to prove our results is (c, t)-track layouts, a combinatorial tool that can be represented as a drawing where: (i) each vertex is assigned to one of t horizontal layers (tracks), (ii) no two adjacent vertices are on the same track, (iii) each edge receives one of c colors, so that no two edges of the same color (u, v) and (w, z) cross if u, w are on the same track, and v, z are on the same track. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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