Abstract
We prove that every n-vertex graph G with path-width pw(G) has a three-dimensional straight-line grid drawing with O(pw(G)2·n) volume. Thus for graphs with bounded path-width the volume is O(n), and it follows that for graphs with bounded tree-width, such as series-parallel graphs, the volume is O(n log2 n). No better bound than O(n 2) was previously known for drawings of series-parallel graphs. For planar graphs we obtain three-dimensional drawings with O(n 2) volume and O(√n) aspect ratio, whereas all previous constructions with O(n 2) volume have Θ(n) aspect ratio.
Research supported by NSERC and FCAR.
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Dujmović, V., Morin, P., Wood, D.R. (2002). Path-Width and Three-Dimensional Straight-Line Grid Drawings of Graphs. In: Goodrich, M.T., Kobourov, S.G. (eds) Graph Drawing. GD 2002. Lecture Notes in Computer Science, vol 2528. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36151-0_5
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