H. de Fraysseix

Bio: H. de Fraysseix is an academic researcher from Centre national de la recherche scientifique. The author has contributed to research in topics: Planar graph & 1-planar graph. The author has an hindex of 3, co-authored 4 publications receiving 908 citations.

How to draw a planar graph on a grid

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.

Representation of planar graphs by segments

Abstract: Note: Professor Pach's number: [100] Reference DCG-CHAPTER-2008-013 Record created on 2008-11-18, modified on 2017-05-12

Connectivity of Planar Graphs

Abstract: We give here three simple linear time algorithms on planar graphs: a 4-connexity test for maximal planar graphs, an algorithm enumerating the triangles and a 3-connexity test. Although all these problems got already linear-time solutions, the presented algorithms are both simple and ecient. They are based on some new theoretical results.

Research Problems in Discrete Geometry

Abstract: Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Embedding planar graphs on the grid

Abstract: We show that each plane graph of order n 2 3 has a straight line embedding on the n-2 by n-2 grid. This embedding is computable in time O(n). A nice feature of the vertex-coordinates is that they have a purely combinatorial meaning.

Appearance-preserving simplification

Abstract: We present a new algorithm for appearance-preserving simplification. Not only does it generate a low-polygon-count approximation of a model, but it also preserves the appearance. This is accomplished for a particular display resolution in the sense that we properly sample the surface position, curvature, and color attributes of the input surface. We convert the input surface to a representation that decouples the sampling of these three attributes, storing the colors and normals in texture and normal maps, respectively. Our simplification algorithm employs a new texture deviation metric, which guarantees that these maps shift by no more than a user-specified number of pixels on the screen. The simplification process filters the surface position, while the runtime system filters the colors and normals on a per-pixel basis. We have applied our simplification technique to several large models achieving significant amounts of simplification with little or no loss in rendering quality. CR Categories: I.3.5: Object hierarchies, I.3.7: Color, shading, shadowing, and texture Additional

Drawing planar graphs using the canonical ordering

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.

On the minimum ropelength of knots and links

Abstract: The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are C 1,1 curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.