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Point-set embeddings of plane 3-trees

Rahnuma Islam Nishat, Debajyoti Mondal, Md. Saidur Rahman
2012 Computational geometry  
We prove an Ω(n log n) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree.  ...  In this paper, we give an O (n 2 ) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists.  ...  Graph Drawing & Information Visualization" supported by the Ministry of Science and Information & Communication Technology, Government of Bangladesh.  ... 
doi:10.1016/j.comgeo.2011.09.002 fatcat:u5tgclkngneqfbzil5aiy3iata

Point-Set Embeddings of Plane 3-Trees [chapter]

Rahnuma Islam Nishat, Debajyoti Mondal, Md. Saidur Rahman
2011 Lecture Notes in Computer Science  
We prove an Ω(n log n) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree.  ...  In this paper, we give an O (n 2 ) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists.  ...  Graph Drawing & Information Visualization" supported by the Ministry of Science and Information & Communication Technology, Government of Bangladesh.  ... 
doi:10.1007/978-3-642-18469-7_29 fatcat:4au3pwcbszawzm35espqszfiiy

On the Hardness of Point-Set Embeddability [chapter]

Stephane Durocher, Debajyoti Mondal
2012 Lecture Notes in Computer Science  
plane 3-trees.  ...  A point-set embedding of a plane graph G with n vertices on a set S of n points is a straight-line drawing of G, where the vertices of G are mapped to distinct points of S.  ...  2-connected plane graph admits a point-set embedding on a given set of points [5] .  ... 
doi:10.1007/978-3-642-28076-4_16 fatcat:tesnzegtijdgzjgierj6bfsfni

Embedding Plane 3-Trees in ℝ2 and ℝ3 [chapter]

Stephane Durocher, Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman, Sue Whitesides
2012 Lecture Notes in Computer Science  
In this paper, we give an O(n 4/3+ )expected time algorithm to decide whether a plane 3-tree with n vertices admits a point-set embedding on a given set of n points in general position in R 2 and compute  ...  We also prove that given a plane 3-tree G with n vertices, a set P of n points in R 3 that are not necessarily in general position and a mapping of the three outer vertices of G to three different points  ...  Point-Set Embeddings of Plane 3-Trees in R 3 Given a plane 3-tree G with n vertices, a set P of n points (not necessarily in general position) in R 2 and a mapping for the outer vertices of G to three  ... 
doi:10.1007/978-3-642-25878-7_5 fatcat:kqhbp346njdkvhpuapjhq7otsu

Improved Algorithms for the Point-Set Embeddability problem for Plane 3-Trees [article]

Tanaeem M. Moosa, M. Sohel Rahman
2010 arXiv   pre-print
In the point set embeddability problem, we are given a plane graph G with n vertices and a point set S with n points.  ...  Recently, in DBLP:conf/gd/NishatMR10, a complete characterization for this problem on a special class of graphs known as the plane 3-trees was presented along with an efficient algorithm to solve the problem  ...  A plane 3-tree of 17 vertices Fig. 2 . 2 The representative tree of the plane 3-tree of Fig. 3 . 3 An example of point set embedding of a plane 3-tree.Example 1 InFigure 3(a) we present an example  ... 
arXiv:1012.0230v1 fatcat:s3nj7txthrbypfepwidhum3nam

Complexity of Planar Embeddability of Trees inside Simple Polygons [article]

Alireza Bagheri, Mohammadreza Razzazi
2009 arXiv   pre-print
Geometric embedding of graphs in a point set in the plane is a well known problem.  ...  Given a point set in the plane bounded by a simple polygon and a free tree, we show that deciding whether there is a planar straight-line embedding of the tree on the point set inside the simple polygon  ...  Let P be a set of n points in the plane inside a simple polygon S, and let T be a n-node free tree.  ... 
arXiv:0908.4041v1 fatcat:yzotlcqzuzcjvoroiktsi3q52e

On Universal Point Sets for Planar Graphs [article]

Jean Cardinal, Michael Hoffmann, Vincent Kusters
2013 arXiv   pre-print
A set P of points in R^2 is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P.  ...  Finally, we describe a collection G of 7'393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane  ...  Hence not every planar 3-tree on 15 vertices can be embedded on any given set of 15 points.  ... 
arXiv:1209.3594v5 fatcat:qyfi76fkuvcclkcg4eo63e5aje

On Universal Point Sets for Planar Graphs [chapter]

Jean Cardinal, Michael Hoffmann, Vincent Kusters
2013 Lecture Notes in Computer Science  
A set P of points in R 2 is n-universal if every planar graph on n vertices admits a plane straight-line embedding on P .  ...  Finally, we describe a collection G of 7 393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane  ...  Proof: Let P ⊂ R 2 be a set of n points and denote by F n ⊆ T n the set of labeled planar 3-trees from T n that admit a plane straight-line embedding onto P .  ... 
doi:10.1007/978-3-642-45281-9_3 fatcat:aw5pswv7ejcobhaphx7ldetrxa

On Universal Point Sets for Planar Graphs

Jean Cardinal, Michael Hoffmann, Vincent Kusters
2015 Journal of Graph Algorithms and Applications  
A set P of points in R 2 is n-universal if every planar graph on n vertices admits a plane straight-line embedding on P .  ...  Finally, we describe a collection G of 7 393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane  ...  Proof: Let P ⊂ R 2 be a set of n points and denote by F n ⊆ T n the set of labeled planar 3-trees from T n that admit a plane straight-line embedding onto P .  ... 
doi:10.7155/jgaa.00374 fatcat:ehvzq7d4ezb3dpzpik7a6rbk4y

Straight-Line Embeddings of Two Rooted Trees in the Plane

A. Kaneko, M. Kano
1999 Discrete & Computational Geometry  
We prove the following theorem: Let T 1 and T 2 be two disjoint rooted trees with roots v 1 and v 2 , respectively, and let P be a set of |T 1 ∪ T 2 | points in the plane in general position containing  ...  Moreover, we give a O(n 2 log n) time algorithm for finding such an embedding, where n is the number of vertices contained in T 1 ∪ T 2 .  ...  Acknowledgments The authors would like to thank Hitoshi Suzuki for his helpful suggestions on the complexities of algorithms. They would also like to thank the referees for their helpful comments.  ... 
doi:10.1007/pl00009441 fatcat:rqsirivg2nclhol4wjdux2ggcq

On embedding an outer-planar graph in a point set [chapter]

Prosenjit Bose
1997 Lecture Notes in Computer Science  
Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(nlog s n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm  ...  If the dual of G is a path, then an optimal O(n log n) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.  ...  Acknowledgements: The author wishes to thank Kilani Ghoudi for helpful discussions on this topic, and Janos Pach for pointing out some missing references.  ... 
doi:10.1007/3-540-63938-1_47 fatcat:ojzi4kaw4fetpawt72coecrqd4

Universal Line-Sets for Drawing Planar 3-Trees [chapter]

Md. Iqbal Hossain, Debajyoti Mondal, Md. Saidur Rahman, Sammi Abida Salma
2012 Lecture Notes in Computer Science  
In this paper we show that a set of n−3 2 + 3 parallel lines or a set of n+3 4 concentric circles are universal for drawing planar 3-trees with n vertices.  ...  A by-product of our algorithm is the generalization of the known bijection between plane 3-trees and rooted full ternary trees to the bijection between planar 3-trees and unrooted full ternary trees.  ...  Acknowledgment We thank the Ministry of Science and Information & Communication Technology, Government of Bangladesh for supporting the project "Facility Upgradation for Sustainable Research on Graph Drawing  ... 
doi:10.1007/978-3-642-28076-4_15 fatcat:oeaxafsnqbfnhlphusp5pijtlm

Universal Line-Sets for Drawing Planar 3-Trees

Md. Iqbal Hossain, Debajyoti Mondal, Md. Saidur Rahman, Sammi Abida Salma
2013 Journal of Graph Algorithms and Applications  
In this paper we show that a set of n−3 2 + 3 parallel lines or a set of n+3 4 concentric circles are universal for drawing planar 3-trees with n vertices.  ...  A by-product of our algorithm is the generalization of the known bijection between plane 3-trees and rooted full ternary trees to the bijection between planar 3-trees and unrooted full ternary trees.  ...  Acknowledgment We thank the Ministry of Science and Information & Communication Technology, Government of Bangladesh for supporting the project "Facility Upgradation for Sustainable Research on Graph Drawing  ... 
doi:10.7155/jgaa.00285 fatcat:nucchstwcjd3nb4zgmcy4kli6e

The Connectivities of Leaf Graphs of Sets of Points in the Plane

Atsushi KANEKO, Kiyoshi YOSHIMOTO
2001 Tokyo Journal of Mathematics  
Let U be a set of n points in the plane. If no three points are collinear, then we s a y  ...  Glicia, A note on the tree g r aph of a set of points in the plane, Proc. of 9th Canadian conference on comput. geom., (1997), 46-49 [3] Y. Ikebe, M. Perles, A. Tamura and S.  ...  [3] showed that any rooted tree with n vertices can be embedded as a non-selntersecting tree on a given set U, the root being mapped to an arbitrary specied point of U. Furthermore A.  ... 
doi:10.3836/tjm/1255958194 fatcat:fwgv6okcfrgkvjlg754elvxftm

Approximating minimum Steiner point trees in Minkowski planes

M. Brazil, C. J. Ras, D. A. Thomas
2010 Networks  
Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number  ...  We also introduce a new canonical form for minimum Steiner point trees in the Euclidean plane; this demonstrates that minimum Steiner point trees are shortest total length trees with a certain discrete-edge-length  ...  The authors wish to thank Jamie Evans for partaking in many fruitful discussions during the development of this paper. We would also like to thank the referees for their insightful comments.  ... 
doi:10.1002/net.20376 fatcat:f3wpqkvkojhexh75kr3h6w5hbu
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