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Graph drawing [chapter]

Roberto Tamassia, Giuseppe Liotta
2004 Handbook of Discrete and Computational Geometry, Second Edition  
Universal lower bounds on the angular resolution exist that depend only on the degree of the graph.  ...  O(n) O(1) 114, 81] degree-4 tree orthogonal grid with leaves on convex hull O(n log n) O(1) 11] Table 1 . 5 : 15 Universal asymptotic upper bounds for the area and lower bounds for the angular  ... 
doi:10.1201/9781420035315.ch52 fatcat:suzr5ked2veg7pmtmgpfppvrua

Advances in the theory and practice of graph drawing

Roberto Tamassia
1999 Theoretical Computer Science  
Specific topics include bounds and tradeoffs for drawing properties. tlrlce-dimensional Icpt-ebentations, methods fur constraint satisfaction, and enptlr imental studieb.  ...  Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph drawing.  ...  Universal lower bounds on the angular resolution exist that depend only on the degree of the graph.  ... 
doi:10.1016/s0304-3975(98)00272-2 fatcat:2ctscsnpxrg6zbu2cmcgo3raoi

Bandwidth, treewidth, separators, expansion, and universality

Julia Böttcher, Klaas P. Pruessmann, Anusch Taraz, Andreas Würfl
2008 Electronic Notes in Discrete Mathematics  
As a consequence for each γ > 0 every n-vertex graph with minimum degree ( 3 4 + γ)n contains a copy of every bounded-degree planar graph on n vertices.  ...  We prove that planar graphs with bounded maximum degree have sublinear bandwidth.  ...  1 Bandwidth and universality for planar graphs Let G = (V, E) be a graph on n = |V | vertices.  ... 
doi:10.1016/j.endm.2008.06.018 fatcat:dk73isgg5vgsdoy5lkvzgdtmli

Universal Graphs for Bounded-Degree Trees and Planar Graphs

Sandeep N. Bhatt, F. R. K. Chung, F. T. Leighton, Arnold L. Rosenberg
1989 SIAM Journal on Discrete Mathematics  
Universal graphs with n vertices and O(n log n) edges are also constructed that contain all bounded-degree planar graphs on n vertices as subgraphs.  ...  How small can a graph be that contains as subgraphs all trees on n vertices with maximum degree d?  ...  While the universal graph for bounded-degree trees is optimal (within a constant factor), the universal graph for bounded-degree planar  ... 
doi:10.1137/0402014 fatcat:3xzyqrsqifdwbbt2sh7ydvjhem

Oriented graph coloring

Eric Sopena
2001 Discrete Mathematics  
We survey the main results that have been obtained on oriented graph colorings.  ...  An oriented k-coloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset  ...  For small values of k, we know that the oriented chromatic number of every graph with maximum degree two is at most 5 (this bound is tight since the directed cycle on ÿve vertices has oriented chromatic  ... 
doi:10.1016/s0012-365x(00)00216-8 fatcat:qozv2mq72fgvlj2aur2lxib2aa

Locally recoverable codes from planar graphs

Kathryn Haymaker, Justin O'Pella
2019 Journal of Algebra Combinatorics Discrete Structures and Applications  
We prove that the minimum distance of planar graph LRCs is equal to the girth of the graph, and we also establish a new bound on the rate of planar graph edge codes.  ...  In this paper we apply Kadhe and Calderbank's definition of LRCs from convex polyhedra and planar graphs [4] to analyze the codes resulting from 3-connected regular and almost regular planar graphs.  ...  Acknowledgment: The authors would like to thank the referees for their helpful suggestions.  ... 
doi:10.13069/jacodesmath.645021 fatcat:j6k42ueqtngn3pxujok2lcmwxu

Succinct Representations of Separable Graphs [chapter]

Guy E. Blelloch, Arash Farzan
2010 Lecture Notes in Computer Science  
In particular, we show constanttime support for adjacency, degree and neighborhood queries.  ...  Separable graphs are those that admit a O(n c )-separator theorem where c < 1. Many graphs that arise in practice are indeed separable. For instance, graphs with a bounded genus are separable.  ...  To show support for adjacency queries in section 3.2, we first oriented the edges of the graph such that each vertex has a bounded out-degree (lemma 5).  ... 
doi:10.1007/978-3-642-13509-5_13 fatcat:b56hksac65awvlf5nwtljnujva

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

David Eppstein
2014 Journal of Graph Algorithms and Applications  
Drawing line arrangement graphs is the main task in the Planarity puzzle.  ...  We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^7/6).  ...  Subquadratic bounds are known on the size of universal point sets for subclasses of the planar graphs including the outerplanar graphs [14] , simply-nested planar graphs [1, 2] , planar 3-trees [11]  ... 
doi:10.7155/jgaa.00319 fatcat:gtq6jgfhjzg6rijsf3r6rghm2e

Planar and Plane Slope Number of Partial 2-Trees [chapter]

William Lenhart, Giuseppe Liotta, Debajyoti Mondal, Rahnuma Islam Nishat
2013 Lecture Notes in Computer Science  
We prove tight bounds (up to a small multiplicative or additive constant) for the plane and the planar slope numbers of partial 2-trees of bounded degree.  ...  As a byproduct of our techniques, we answer a long standing question by Garg and Tamassia about the angular resolution of the planar straight-line drawings of series-parallel graphs of bounded degree.  ...  In this paper we prove tight bounds (up to a small multiplicative or additive constant) for the plane and the planar slope numbers of planar 2-trees of bounded degree.  ... 
doi:10.1007/978-3-319-03841-4_36 fatcat:aizqfijgnjfajgqz6b2irafuaa

Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs [article]

Julia Böttcher, Klaas P. Pruessmann, Anusch Taraz, Andreas Würfl
2009 arXiv   pre-print
a copy of every bounded-degree planar graph on n vertices if n is sufficiently large.  ...  As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each gamma>0 every n-vertex graph with minimum degree ((3/4)+gamma)n contains  ...  Universal graphs for bounded degree planar graphs have also been studied in [3, 6] . Corollary 19.  ... 
arXiv:0910.3014v1 fatcat:7jkq57jrurc2hcum4wol2ff6cy

Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs

Julia Böttcher, Klaas P. Pruessmann, Anusch Taraz, Andreas Würfl
2010 European journal of combinatorics (Print)  
of every bounded-degree planar graph on n vertices if n is sufficiently large.  ...  As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each γ > 0 every n-vertex graph with minimum degree ( 3 4 + γ )n contains a copy  ...  Acknowledgements The first author would like to thank David Wood for fruitful discussions in an early stage of this project. In addition we thank two anonymous referees for their helpful suggestions.  ... 
doi:10.1016/j.ejc.2009.10.010 fatcat:p3ldquvirrfq3ev2d5puafgddi

Processing Huge Graphs with Stxxl

Roman Dementiev
2006 Electronic Notes in Discrete Mathematics  
) q Batched Smallest Degree Last (BSDL) q BSDL: Planar Graphs q Experiments: Small Inputs q Experiments: Large Inputs q Experiments: EM7 on Huge Planar Graphs q Summary University of Karlsruhe, Roman Dementiev  ...  First (HDF) q Smallest Degree Last (SDL) q Batched Smallest Degree Last (BSDL) q BSDL: Planar Graphs q Experiments: Small Inputs q Experiments: Large Inputs q Experiments: EM7 on Huge Planar Graphs q  ... 
doi:10.1016/j.endm.2006.06.060 fatcat:m2cyynqznzaztbtlt45jebt7vy

A Note On Universal Point Sets for Planar Graphs [article]

Manfred Scheucher and Hendrik Schrezenmaier and Raphael Steiner
2019 arXiv   pre-print
universal sets for n > 15.  ...  This strengthens the result by Cardinal, Hoffmann, and Kusters (2015), that all planar graphs on n < 10 vertices can be simultaneously drawn on particular 'universal' sets of n points while there are no  ...  A point set, which is n-universal for the family of all planar graphs, is simply called n-universal.  ... 
arXiv:1811.06482v3 fatcat:s7x2wkd5w5ggpjdkws4szxloqy

Bidimensional Structures: Algorithms, Combinatorics and Logic (Dagstuhl Seminar 13121)

Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, Dimitrios M. Thilikos, Marc Herbstritt
2013 Dagstuhl Reports  
Then a dynamic programming algorithm for Planar Completion to Bounded Diameter on graphs of bound treewidth would result in the construction of the desired algorithm.  ...  More generally, how can we design bounded-treewidth dynamic-programming algorithms for planar completion problems?  ...  Disjoint Paths in Planar Graphs Daniel Lokshtanov, University of Bergen, daniello@ii.uib.no For the k disjoint paths problem in planar graphs, is there a fixed-parameter algorithm with running time 2 k  ... 
doi:10.4230/dagrep.3.3.51 dblp:journals/dagstuhl-reports/DemaineFHT13 fatcat:vbtdm45hovds5eyorghjbz4kr4

Page 5749 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
{For the entire collection see MR 2003d:00011.} 2003h:05107 05C35 05C10 68R10 Capalbo, Michael (1-JHOP-MS; Baltimore, MD) Small universal graphs for bounded-degree planar graphs.  ...  The author 05C Graph theory 2003h:05110 constructs a universal graph H for the family of planar graphs with at most N vertices and maximum degree at most k.  ... 
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