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### Optimal algorithms to embed trees in a point set [chapter]

Prosenjit Bose, Michael McAllister, Jack Snoeyink
1996 Lecture Notes in Computer Science
In the degree-constrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2n − 2 and are asked to embed a tree in P using straight  ...  In the rootedtree embedding problem we are given a rooted tree T with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the root  ...  Acknowledgments We thank the referees for their suggestions on improving the presentation and organisation of this paper.  ...

### Optimal Algorithms to Embed Trees in a Point Set

P. Bose, M. McAllister, J. Snoeyink
1997 Journal of Graph Algorithms and Applications
In the degree-constrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2n − 2 and are asked to embed a tree in P using straight  ...  In the rootedtree embedding problem we are given a rooted tree T with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the root  ...  Acknowledgments We thank the referees for their suggestions on improving the presentation and organisation of this paper.  ...

### Optimal Algorithms to Embed Trees in a Point Set [chapter]

Prosenjit Bose, Michael McAllister, Jack Snoeyink
2002 Graph Algorithms and Applications I
In the degree-constrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2n − 2 and are asked to embed a tree in P using straight  ...  In the rootedtree embedding problem we are given a rooted tree T with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the root  ...  Acknowledgments We thank the referees for their suggestions on improving the presentation and organisation of this paper.  ...

### Planar embedding of trees on point sets without the general position assumption

Asghar ASGHARIAN SARDROUD, Alireza BAGHERI
2015 Turkish Journal of Mathematics
We also give some results when the problem is limited to degree-constrained trees and point sets having constant number of collinear points.  ...  The problem of point-set embedding of a planar graph G on a point set P in the plane is defined as finding a straight-line planar drawing of G such that the nodes of G are mapped one to one on the points  ...  Therefore, the complexity of the problem is still open if we limit the problem to the degree-constrained trees or limit the maximum number of collinear points in the point set to a constant.  ...

### Page 1921 of Mathematical Reviews Vol. , Issue 97C [page]

1997 Mathematical Reviews
In the degree- constrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2m — 2 and are asked to find an embedding of a tree in  ...  In the rooted-tree embedding problem we are given a rooted-tree JT with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the  ...

### Computing Steiner points for gradient-constrained minimum networks

D.A. Thomas, J.F. Weng
2010 Discrete Optimization
A degree-3 Steiner point is labelled minimal if the total length of the three adjacent edges is minimized for a given labelling.  ...  Let T g be a gradient-constrained minimum network, that is, a minimum length network spanning a given point set in 3-dimensional space with edges that are constrained to have gradients no more than an  ...  The Steiner points in LMGCNs can be classified into different types, called labellings, according to whether the gradients of their incident edges are less than, equal to or greater than m when embedded  ...

### Semi-bipartite Graph Visualization for Gene Ontology Networks [chapter]

Kai Xu, Rohan Williams, Seok-Hee Hong, Qing Liu, Ji Zhang
2010 Lecture Notes in Computer Science
In this paper we present algorithms which test in polynomial time if a given hypergraph H has a planar support that is (i) a path, (ii) a cycle, (iii) a tree, or (iv) a tree where the maximal degree of  ...  We also show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most 2 O(∆) . In particular, we answer the question of Dujmović et al.  ...

### Overlay addressing and routing system based on hyperbolic geometry

Cyril Cassagnes, Telesphore Tiendrebeogo, David Bromberg, Damien Magoni
2011 2011 IEEE Symposium on Computers and Communications (ISCC)
Local knowledge routing schemes based on virtual coordinates taken from the hyperbolic plane have attracted considerable interest in recent years.  ...  In this paper, we propose a new approach for seizing the power of the hyperbolic geometry.  ...  This particular tiling splits the hyperbolic plane in distinct spaces and constructs our embedded tree.  ...

### 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.  ...  Acknowledgements This work is done in Graph Drawing & Information Visualization Laboratory of the Department of CSE, BUET established under the project "Facility Upgradation for Sustainable Research on  ...

### Polynomial area bounds for MST embeddings of trees

Fabrizio Frati, Michael Kaufmann
2011 Computational geometry
In their seminal paper on geometric minimum spanning trees, Monma and Suri (1992) [31] showed how to embed any tree of maximum degree 5 as a minimum spanning tree in the Euclidean plane.  ...  In this paper, we show how to construct MST embeddings of arbitrary trees of maximum degree 3 and 4 within polynomial area.  ...  The initial results on complete binary trees were achieved together with Roberto Tamassia. A preliminary and short version of this paper appeared in [21] .  ...

### 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.  ...  Acknowledgements This work is done in Graph Drawing & Information Visualization Laboratory of the Department of CSE, BUET established under the project "Facility Upgradation for Sustainable Research on  ...

### A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem [article]

Thomas Bläsius, Ignaz Rutter
2015 arXiv   pre-print
We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases.  ...  Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices.  ...  Conclusion In this paper we have introduced the cd-tree and we have shown that it can be used to reformulate the classic c-planarity problem as a constrained embedding problem.  ...

### Point-Set Embedding of Trees with Edge Constraints [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer, Stephen Wismath
2008 Lecture Notes in Computer Science
Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S.  ...  We concentrate on trees and show how to compute the output in O(n 2 log n) time and with at most 1 + 2 k/2 bends per edge, where k is the number of vertices of the given subdrawing.  ...  Introduction Let G be a planar graph with n vertices and let S be a set of n points in the plane.  ...

### Fast geometric approximation techniques and geometric embedding problems

Marshall W. Bern, Howard J. Karloff, Prabhakar Raghavan, Baruch Schieber
1992 Theoretical Computer Science
We give the example of approximating the convex hull of a set of points in the plane . ..' pn > of II points in R", and an undirected graph G = (V, E) on n vertices.  ...  We give fast approximation algorithms for embedding these graphs on the line and in the plane in several metrics.  ...  Trees Our EMST approximation algorithm yields an approximation algorithm for embedding bounded-degree trees, such as a complete binary tree, onto points P in the plane.  ...

### Optimal Orthogonal Drawings of Planar 3-Graphs in Linear Time [article]

Walter Didimo and Giuseppe Liotta and Giacomo Ortali and Maurizio Patrignani
2019 arXiv   pre-print
., a planar graph with vertex degree at most three), what is the best computational upper bound to compute a bend-minimum planar orthogonal drawing of G in the variable embedding setting?  ...  In this setting the algorithm can choose among the exponentially many planar embeddings of G the one that leads to an orthogonal drawing with the minimum number of bends.  ...  SPQR-Trees. Let G be a biconnected graph. An SPQR-tree T of G represents the decomposition of G into its triconnected components and can be computed in linear time [8, 17, 19] .  ...
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