Point-set embeddings of plane 3-trees

Nishat, Rahnuma Islam; Mondal, Debajyoti; Rahman, Md. Saidur

doi:10.1016/j.comgeo.2011.09.002

Cited by 14 publications

(23 citation statements)

References 11 publications

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“…We prove this with the following two lemmas. We do not claim originality for the proofs: ideas similar to Lemma 5 appeared in [26] and a proof for Lemma 6 in a slightly different setting appeared in [27].…”

Section: Simultaneous Geometric Embeddingsmentioning

confidence: 99%

On Universal Point Sets for Planar Graphs

Cardinal¹,

Hoffmann²,

Kusters³

2015

JGAA

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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 . Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n ≥ 15. Conversely, we use a computer program to show that there exist universal point sets for all n ≤ 10 and to enumerate all corresponding order types. 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 straight-line embedding of all graphs in G.

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Section: Simultaneous Geometric Embeddingsmentioning

confidence: 99%

On Universal Point Sets for Planar Graphs

Cardinal¹,

Hoffmann²,

Kusters³

2015

JGAA

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“…In this paper, we follow up the work of [15] and improve upon their result from an algorithmic point of view. In [15], Nishat et al presented an O(n 2 log n) time algorithm that can decide whether a plane 3-tree G of n vertices admits a point-set embedding on a given set of n points or not and compute a point-set embedding of G if such an embedding exists. In this paper, we show how to improve the running time of the above algorithm.…”

Section: Introductionmentioning

confidence: 98%

“…Plane 3-trees belong to an interesting class of graphs and recently a number of different drawing algorithms have been presented in the literature on plane 3-trees [2,14,15]. In this paper, we follow up the work of [15] and improve upon their result from an algorithmic point of view. In [15], Nishat et al presented an O(n 2 log n) time algorithm that can decide whether a plane 3-tree G of n vertices admits a point-set embedding on a given set of n points or not and compute a point-set embedding of G if such an embedding exists.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, Nishat et al [15] studied the point set embeddability problem on a class of graphs known as the plane 3-tree. Plane 3-trees belong to an interesting class of graphs and recently a number of different drawing algorithms have been presented in the literature on plane 3-trees [2,14,15]. In this paper, we follow up the work of [15] and improve upon their result from an algorithmic point of view.…”

Section: Introductionmentioning

confidence: 99%

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Improved Algorithms for the Point-Set Embeddability Problem for Plane 3-Trees

Moosa

Rahman

2012

Discrete Math. Algorithm. Appl.

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In the point set embeddability problem, we are given a plane graph G with n vertices and a point set S with n points. Now the goal is to answer the question whether there exists a straight-line drawing of G such that each vertex is represented as a distinct point of S as well as to provide an embedding if one does exist. Recently, in [15], 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. In this paper, we use the same characterization to devise an improved algorithm for the same problem. Much of the efficiency we achieve comes from clever uses of the triangular range search technique. We also study a generalized version of the problem and present improved algorithms for this version of the problem as well.

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“…The problem is known to be NP-hard [9], and remains NP-hard even for 3-connected planar graphs [17], triangulations and 2-connected outerplanar graphs [4]. However, it has a polynomial-time solution for 3-trees [19,28,18]. In a polyline embedding of a plane graph, the edges are represented by pairwise noncrossing polygonal paths.…”

mentioning

confidence: 99%