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Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
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Cited by 36 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
Planar graphs, via wellorderly maps and trees
 IN 30 TH INTERNATIONAL WORKSHOP, GRAPH  THEORETIC CONCEPTS IN COMPUTER SCIENCE (WG), VOLUME 3353 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of th ..."
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Cited by 18 (4 self)
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The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worstcase, a rate of 4.91 bits per node and of 2.82 bits per edge.
Convex drawings of 3connected plane graphs
 Algorithmica
, 2007
"... We use Schnyder woods of 3connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0 ≤ ∆ ≤ n 2 − 2. The algorithm is a refinement of the fac ..."
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Cited by 10 (0 self)
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We use Schnyder woods of 3connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0 ≤ ∆ ≤ n 2 − 2. The algorithm is a refinement of the facecountingalgorithm, thus, in particular, the size of the grid is at most (f − 2) × (f − 2). The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder’s and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3connected planar graphs. The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to 7 7
Geodesic Embeddings and Planar Graphs
, 2002
"... Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better u ..."
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Cited by 9 (6 self)
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Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better understanding of the connections between these objects. In this article we prove a characterization of 3connected planar graphs as those graphs admitting rigid geodesic embeddings, a bijection between Schnyder labelings and rigid geodesic embeddings, a strong version of the BrightwellTrotter theorem.
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
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Cited by 6 (1 self)
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Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �
Some Applications of Orderly Spanning Trees in Graph Drawing
 IN PROCEEDINGS OF THE 10TH INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING, LNCS 2528
, 2002
"... Orderly spanning trees seem to have the potential of becoming a new and promising technique capable of unifying known results as well as deriving new results in graph drawing. Our exploration in this paper provides new evidence to demonstrate such a potential. Two applications of the orderly spa ..."
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Cited by 2 (2 self)
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Orderly spanning trees seem to have the potential of becoming a new and promising technique capable of unifying known results as well as deriving new results in graph drawing. Our exploration in this paper provides new evidence to demonstrate such a potential. Two applications of the orderly spanning trees of plane graphs are investigated. Our first
270 Planar Graphs, via WellOrderly Maps and Trees
"... Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n),whereα ≈ 4.91. A direct consequenc ..."
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Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n),whereα ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worstcase, a rate of 4.91 bits per node and of 2.82 bits per edge. 1