Topic

# Slope number

About: Slope number is a research topic. Over the lifetime, 251 publications have been published within this topic receiving 7873 citations.

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23 Jul 1998

TL;DR: In this paper, the authors describe fundamental algorithmic techniques for constructing drawings of graphs and provide an accurate, accessible reflection of the rapidly expanding field of graph drawing, using a reference manual.

Abstract: From the Publisher:
This book is designed to describe fundamental algorithmic techniques for constructing drawings of graphs. Suitable as a book or reference manual, its chapters offer an accurate, accessible reflection of the rapidly expanding field of graph drawing.

1,754 citations

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TL;DR: It is shown that any setF, which can support a Fáry embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

Abstract: Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fary embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fary embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

755 citations

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Brown University

^{1}TL;DR: It is shown that upward planarity testing and rectilinear planar testing are NP-complete problems and that it is NP-hard to approximate the minimum number of bends in a planar orthogonal drawing of an n-vertex graph with an $O(n^{1-\epsilon})$ error for any $\ep silon > 0$.

Abstract: A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction and no two edges cross An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment and no two edges cross Testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures For example, upward planarity is useful for the display of order diagrams and subroutine-call graphs, while rectilinear planarity is useful for the display of circuit schematics and entity-relationship diagrams
We show that upward planarity testing and rectilinear planarity testing are NP-complete problems We also show that it is NP-hard to approximate the minimum number of bends in a planar orthogonal drawing of an n-vertex graph with an $O(n^{1-\epsilon})$ error for any $\epsilon > 0$

363 citations

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TL;DR: This work proposes a linear-time algorithm, a variant of one by Otten and van Wijk, that generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way.

Abstract: We propose a linear-time algorithm for generating a planar layout of a planar graph. Each vertex is represented by a horizontal line segment and each edge by a vertical line segment. All endpoints of the segments have integer coordinates. The total space occupied by the layout is at mostn by at most 2n---4. Our algorithm, a variant of one by Otten and van Wijk, generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way. The algorithm is based on the concept of abipolar orientation. We discuss relationships among the bipolar orientations of a planar graph.

335 citations

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TL;DR: This work introduces a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid using a new type of ordering on the vertices and faces of triconnected planar graphs.

Abstract: We introduce a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear-time-and-space algorithms can be designed for many graph-drawing problems. Our main results are as follows:
Every triconnected planar graphG admits a planar convex grid drawing with straight lines on a (2n−4)×(n−2) grid, wheren is the number of vertices.
Every triconnected planar graph with maximum degree 4 admits a planar orthogonal grid drawing on ann×n grid with at most [3n/2]+4 bends, and ifn>6, then every edge has at most two bends.
Every planar graph with maximum degree 3 admits a planar orthogonal grid drawing with at most [n/2]+1 bends on an [n/2]×[n/2] grid.
Every triconnected planar graphG admits a planar polyline grid drawing on a (2n−6)×(3n−9) grid with minimum angle larger than 2/d radians and at most 5n−15 bends, withd the maximum degree.

309 citations