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Homotopy height, grid-major height and graph-drawing height [article]

Therese Biedl, Erin Wolf Chambers, David Eppstein, Arnaud De Mesmay,, Tim Ophelders
2019 arXiv   pre-print
But both bounds fall short for some graphs. In this paper, we consider two other parameters, the (simple) homotopy height and the (simple) grid-major height.  ...  It is well-known that both the pathwidth and the outer-planarity of a graph can be used to obtain lower bounds on the height of a planar straight-line drawing of a graph.  ...  k, the graphs with grid-major height k and homotopy-height k can be recognized in linear time.  ... 
arXiv:1908.05706v2 fatcat:ipnmxvhkcnhsrfeyvbnpapu4e4

Page 5652 of Mathematical Reviews Vol. , Issue 2004g [page]

2004 Mathematical Reviews  
of crossings in graph drawings and VLSI layout areas.  ...  Summary: “Preserving the ‘mental map’ is a major goal of interac- tive graph drawing algorithms. Several models have been proposed for formalizing the notion of mental map.  ... 

Applications of Topology to the Analysis of 1-Dimensional Objects (Dagstuhl Seminar 17072)

Benjamin Burton, Maarten Löffler, Carola Wenk, Erin Moriarty Wolf Chambers, Marc Herbstritt
2017 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 17072 "Applications of Topology to the Analysis of 1-Dimensional Objects".  ...  This problem is known as the homotopy height problem, as the sweep encodes a homotopy across the disk.  ...  and computational geometry, all the way to graph drawing.  ... 
doi:10.4230/dagrep.7.2.64 dblp:journals/dagstuhl-reports/BurtonLWC17 fatcat:syifrcehlrgungdoitaxfb7baa

Persistence Diagrams and the Heat Equation Homotopy [article]

Brittany Terese Fasy
2010 arXiv   pre-print
Furthermore, I formulate a homotopy from g to f by applying the heat equation to the difference function g-f.  ...  I look at the diagrams where M is a square, a sphere, a torus, and a Klein bottle.  ...  Suppose C is the graph of f : R → R. We can think of f as the height function on C.  ... 
arXiv:1002.1937v1 fatcat:ov5kbouf3fct7mrzhptfy4kgbm

Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352)

Maartten Löffler, Anna Lubiw, Saul Schleimer, Erin Moriarty Wolf Chambers, Michael Wagner
2019 Dagstuhl Reports  
The seminar participants investigated problems in: knot theory, trajectory analysis, algorithmic topology, computational geometry of curves, and graph drawing, with an emphasis on how low-dimensional structures  ...  This report documents the program and the outcomes of Dagstuhl Seminar 19352 "Computation in Low-Dimensional Geometry and Topology".  ...  This is because solutions to isotopic Fréchet shrink spiraling parts of the curve infinitesimally small and then unspiral them.  ... 
doi:10.4230/dagrep.9.8.84 dblp:journals/dagstuhl-reports/LofflerLSC19 fatcat:jvf6v5zuhngi3igvpucn7miqum

Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs [article]

Therese Biedl, Veronika Irvine
2017 arXiv   pre-print
once around the minor meridian axis and zero times around the major longitude axis.  ...  In this paper, we study a class of graph drawings that arise from bobbin lace patterns.  ...  and zero times around the major longitude axis).  ... 
arXiv:1708.09778v3 fatcat:utdfdpswdjbctpbrfwax7hus2y

On d-Regular Schematization of Embedded Paths [chapter]

Andreas Gemsa, Martin Nöllenburg, Thomas Pajor, Ignaz Rutter
2011 Lecture Notes in Computer Science  
Motivated by drawing route sketches, we consider the d-regular path schematization problem.  ...  ., a route in a road network) and a positive integer d.  ...  that sketch [4] , or in dynamic graph drawing, where each drawing in a sequence of drawings must be similar to its predecessor [5] .  ... 
doi:10.1007/978-3-642-18381-2_22 fatcat:jr3lvfrsk5dm5fkjthb3gcp2uy

On d -regular schematization of embedded paths

Daniel Delling, Andreas Gemsa, Martin Nöllenburg, Thomas Pajor, Ignaz Rutter
2014 Computational geometry  
Motivated by drawing route sketches, we consider the d-regular path schematization problem.  ...  ., a route in a road network) and a positive integer d.  ...  that sketch [4] , or in dynamic graph drawing, where each drawing in a sequence of drawings must be similar to its predecessor [5] .  ... 
doi:10.1016/j.comgeo.2013.10.002 fatcat:5wgom5mr5jcdnl4rtylzpt34wa

Drawing Bobbin Lace Graphs, or, Fundamental Cycles for a Subclass of Periodic Graphs [chapter]

Therese Biedl, Veronika Irvine
2018 Lecture Notes in Computer Science  
once around the minor meridian axis and zero times around the major longitude axis.  ...  In this paper, we study a class of graph drawings that arise from bobbin lace patterns.  ...  and zero times around the major longitude axis).  ... 
doi:10.1007/978-3-319-73915-1_12 fatcat:lzoax7i7fncptgzh6i5nk2po3i

A Parallel, O(n) Algorithm for an Unbiased, Thin Watershed

Théodore Chabardès, Petr Dokládal, Matthieu Faessel, Michel Bilodeau
2022 Image Processing On Line  
Compilation and usage instruction are included in the README.txt file of the archive.  ...  Experimental results show increased performances regarding execution speed and memory consumption.  ...  Acknowledgment The authors wish to thank the reviewers for their thorough analysis of the text and of the (quite a complex) code; and the managing editor Pascal Monasse for his support and help that went  ... 
doi:10.5201/ipol.2022.215 fatcat:ia4jns56qfa6plfdsdnyoqxspm

A Categorical Approach to Contour, Split and Join Trees with Application to Airway Segmentation [chapter]

Andrzej Szymczak
2010 Mathematics and Visualization  
It computes all sublevel components in thick slices of the input image that have simple topology and branching structure.  ...  Contour, split and join trees can be defined as functors acting on the category of scalar fields, whose morphisms are value-preserving functions.  ...  The process of building the element of S from the combinatorial tree representation is straighforward and basically boils down to drawing the graph -a special case of geometric realization of a simplicial  ... 
doi:10.1007/978-3-642-15014-2_17 fatcat:s6mdsvzkzfcy7pwjutuegjvlcu

MAP

Jehoshua Bruck, Jie Gao, Anxiao (Andrew) Jiang
2005 Proceedings of the 11th annual international conference on Mobile computing and networking - MobiCom '05  
It can be represented compactly by a graph whose size is comparable with the complexity of the geometric features (e.g., the number of holes).  ...  The routing scheme is derived through local decisions based on the names of the source and destination nodes and guarantees delivery with reasonable and natural routes.  ...  Acknowledgements: This work was supported in part by the Lee Center for Advanced Networking at the California Institute of Technology, and by NSF grant CCR-TC-0209042.  ... 
doi:10.1145/1080829.1080839 dblp:conf/mobicom/BruckGJ05 fatcat:sk4i3jq3rfdrvaappkb7qpktyq

MAP: Medial axis based geometric routing in sensor networks

Jehoshua Bruck, Jie Gao, Anxiao (Andrew) Jiang
2006 Wireless networks  
It can be represented compactly by a graph whose size is comparable with the complexity of the geometric features (e.g., the number of holes).  ...  The routing scheme is derived through local decisions based on the names of the source and destination nodes and guarantees delivery with reasonable and natural routes.  ...  Acknowledgements: This work was supported in part by the Lee Center for Advanced Networking at the California Institute of Technology, and by NSF grant CCR-TC-0209042.  ... 
doi:10.1007/s11276-006-9857-z fatcat:q4sb6ra6ubaefcs2psh2hdopsi

Describing shapes by geometrical-topological properties of real functions

S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, M. Spagnuolo
2008 ACM Computing Surveys  
All these have been developed over the years with a specific target domain and it is not trivial to systematize this work and understand the links, similarities and differences among the different methods  ...  Differential topology, and specifically Morse theory, provides a suitable setting for formalizing and solving several problems related to shape analysis.  ...  This work has been developed within the CNR research activities ICT-P03 and DG.RSTL.050.004, and with the support of the EU FP6 Network of Excellence AIM@SHAPE, contract number 506766, by the Italian National  ... 
doi:10.1145/1391729.1391731 fatcat:yg6k2lwerfd63nuhhxxypcnzju

Occlusion-Aware Path Planning to Promote Infrared Positioning Accuracy for Autonomous Driving in a Warehouse

Bai Li, Shiqi Tang, Youmin Zhang, Xiang Zhong
2021 Electronics  
In layer one, a homotopic A* path is searched for in the 2D grid map to roughly connect the initial and goal points.  ...  Simulation results in CarSim software show that the derived paths reduce the poor positioning risk and are easily tracked by a controller.  ...  Suppose the infrared emitter is installed at the height of h onto the ego vehicle, one may draw a line from the 3D point (x wp i , y wp i , h) to each infrared receiver and then check if the line is occluded  ... 
doi:10.3390/electronics10243093 fatcat:4cooqagsjremxcirg7rbxfj6ta
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