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11
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 43 (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
, 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),whereα ≈ 4.91. A direct consequence of thi ..."
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Cited by 21 (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),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.
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 18 (1 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 14 (8 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 8 (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
Drawing Planar Graphs with Reduced Height
"... Abstract. A straightline (respectively, polyline) drawing Γ of a planar graph G on a set Lk of k parallel lines is a planar drawing that maps each vertex of G to a distinct point on Lk and each edge of G to a straight line segment (respectively, a polygonal chain with the bends on Lk) between its e ..."
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Abstract. A straightline (respectively, polyline) drawing Γ of a planar graph G on a set Lk of k parallel lines is a planar drawing that maps each vertex of G to a distinct point on Lk and each edge of G to a straight line segment (respectively, a polygonal chain with the bends on Lk) between its endpoints. The height of Γ is k, i.e., the number of lines used in the drawing. In this paper we compute new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every nvertex planar graph with maximum degree ∆, having a simple cycle separator of size λ, admits a polyline drawing with height 4n/9+O(λ∆), where the previously best known bound was 2n/3. Since λ ∈ O(√n), this implies the existence of a drawing of height at most 4n/9 + o(n) for any planar triangulation with ∆ ∈ o(√n). For nvertex planar 3trees, we compute straightline drawings with height 4n/9 +O(1), which improves the previously best known upper bound of n/2. All these results can be viewed as an initial step towards compact drawings of planar triangulations via choosing a suitable embedding of the input graph. 1
Tradeoffs in Planar Polyline Drawings
"... Abstract. Angular resolution, area and the number of bends are some important aesthetic criteria of a polyline drawing. Although tradeoffs among these criteria have been examined over the past decades, many of these tradeoffs are still not known to be optimal. In this paper we give a new technique ..."
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Abstract. Angular resolution, area and the number of bends are some important aesthetic criteria of a polyline drawing. Although tradeoffs among these criteria have been examined over the past decades, many of these tradeoffs are still not known to be optimal. In this paper we give a new technique to compute polyline drawings for planar triangulations. Our algorithm is simple and intuitive, yet implies significant improvement over the known results. We present the first smooth tradeoff between the area and angular resolution for 2bend polyline drawings of any given planar graph. Specifically, for any given nvertex triangulation, our algorithm computes a drawing with angular resolution r/d(v) at each vertex v, and area f(n, r), for any r ∈ (0, 1], where d(v) denotes the degree at v. For r < 0.389 or r> 0.5, f(n, r) is less than the drawing area required by previous algorithms; f(n, r) ranges from 7.12n2 when r ≤ 0.3 to 32.12n2 when r = 1. 1
Drawing Plane Triangulations with Few Segments
"... Dujmović, Eppstein, Suderman, and Wood showed that every 3connected plane graph G with n vertices admits a straightline drawing with at most 2.5n − 3 segments, which is also the best known upper bound when restricted to plane triangulations. On the other hand, they showed that there exist trian ..."
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Dujmović, Eppstein, Suderman, and Wood showed that every 3connected plane graph G with n vertices admits a straightline drawing with at most 2.5n − 3 segments, which is also the best known upper bound when restricted to plane triangulations. On the other hand, they showed that there exist triangulations requiring 2n − 6 segments. In this paper we show that every plane triangulation admits a straightline drawing with at most (7n − 2∆0 − 10)/3 ≤ 2.33n segments, where ∆0 is the number of cyclic faces in the minimum realizer of G. If the input triangulation is 4connected, then our algorithm computes a drawing with at most (9n − 9)/4 ≤ 2.25n segments. For general plane graphs with n vertices and m edges, our algorithm requires at most (16n − 3m − 28)/3 ≤ 5.33n −m segments, which is smaller than 2.5n − 3 for all m ≥ 2.84n.