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Canonical Ordering Trees and Their Applications in Graph Drawing

Huaming Zhang, Xin He
2005 Discrete & Computational Geometry  
We study the properties of Schnyder's realizers and canonical ordering trees of plane graphs.  ...  Based on these properties, we show that every such a graph has a canonical ordering tree with at most (n + 1)/2 leaves. This improves the previously known bound of (2n + 1)/3 .  ...  We obtained a canonical ordering tree for a 4-connected plane triangulation with at most (n + 1)/2 leaves.  ... 
doi:10.1007/s00454-004-1154-y fatcat:qkwireg74badth4pfap2ypvfjm

Bitonic st-orderings of Biconnected Planar Graphs [chapter]

Martin Gronemann
2014 Lecture Notes in Computer Science  
Vertex orderings play an important role in the design of graph drawing algorithms. Compared to canonical orderings, st-orderings lack a certain property that is required by many drawing methods.  ...  We describe a linear-time algorithm to obtain such an ordering and demonstrate its capabilities with two applications.  ...  Conclusion We have shown that every biconnected planar graph has a bitonic st-order that can be obtained in linear time.  ... 
doi:10.1007/978-3-662-45803-7_14 fatcat:rabmib6p5jdwtcouuvlghen3qe

Canonical Orders and Schnyder Realizers [chapter]

Stephen Kobourov
2015 Encyclopedia of Algorithms  
Wagner in 1936 [12], Fáry in 1948, and Stein in 1951 [10] proved independently that every planar graph has a straight-line drawing.  ...  These two seemingly very different combinatorial structures turn out to be closely related and have since been used in many different problems and in many applications.  ...  Equivalency of Canonical Orders and Schnyder Realizers Here we show that canonical orders and Schnyder realizers, are in fact equivalent, in the sense that a canonical order of a graph defines a Schnyder  ... 
doi:10.1007/978-3-642-27848-8_650-1 fatcat:fj2lghzrdfhnpei7fvvgmfjswu

10 Reasons to Get Interested in Graph Drawing [chapter]

Carla Binucci, Ulrik Brandes, Tim Dwyer, Martin Gronemann, Reinhard von Hanxleden, Marc van Kreveld, Petra Mutzel, Marcus Schaefer, Falk Schreiber, Bettina Speckmann
2019 Lecture Notes in Computer Science  
A selection of results demonstrates the influence of graph drawing on other areas and vice versa.  ...  It encompasses basic research such as graph theory, complexity theory, data structures, and graph algorithms as well as applied research such as software libraries, implementations, and applications.  ...  They also show that a canonical ordering induces one in the dual of a triconnected planar graph. Kant's definition has found numerous applications in graph drawing.  ... 
doi:10.1007/978-3-319-91908-9_6 fatcat:wchqxlk4fbfqdmhdjhhks74joe

Canonical Decomposition, Realizer, Schnyder Labeling and Orderly Spanning Trees of Plane Graphs [chapter]

Kazuyuki Miura, Machiko Azuma, Takao Nishizeki
2004 Lecture Notes in Computer Science  
A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph play an important role in straight-line grid drawings, convex grid drawings, floor-plannings, graph  ...  In this paper, we present a necessary and sufficient condition for their existence, and show that a canonical decomposition, a realizer, a Schnyder labeling, an orderly spanning tree, and an outer triangular  ...  Introduction Recently automatic aesthetic drawing of graphs has created intense interest due to their broad applications, and as a consequence, a number of drawing methods have come out [3-9, 11-13, 15  ... 
doi:10.1007/978-3-540-27798-9_34 fatcat:rc6ux56hdzh2rfkivijeyhakqq

Drawing Planar Graphs with Few Geometric Primitives

Gregor Hültenschmidt, Philipp Kindermann, Wouter Meulemans, André Schulz
2018 Journal of Graph Algorithms and Applications  
We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n - 11)/3 arcs.  ...  In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let n denote the number of vertices of a graph.  ...  We order the heavy paths nondecreasingly by their depth and then draw their subtrees in this order.  ... 
doi:10.7155/jgaa.00473 fatcat:yfl4babp25bsrps5vzr6zd4ggm

CANONICAL DECOMPOSITION, REALIZER, SCHNYDER LABELING AND ORDERLY SPANNING TREES OF PLANE GRAPHS

KAZUYUKI MIURA, MACHIKO AZUMA, TAKAO NISHIZEKI
2005 International Journal of Foundations of Computer Science  
A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph play an important role in straight-line grid drawings, convex grid drawings, floor-plannings, graph  ...  In this paper, we present a necessary and sufficient condition for their existence, and show that a canonical decomposition, a realizer, a Schnyder labeling, an orderly spanning tree, and an outer triangular  ...  Introduction Recently automatic aesthetic drawing of graphs has created intense interest due to their broad applications, and as a consequence, a number of drawing methods have come out [3-9, 11-13, 15  ... 
doi:10.1142/s0129054105002905 fatcat:xnxog6szmfcsjox3nkxpeqlbya

A More Compact Visibility Representation

Goos Kant
1997 International journal of computational geometry and applications  
See Figure 4 for the 4-block tree of the graph in Figure 1 . We root T at an arbitrary node b. We compute a canonical ordering for G b , as de ned in Theorem 2.1, and direct the edges accordingly.  ...  The requirement of using a small area seems to become a core area in the research eld of graph drawing, due to the important applications in VLSI-design and chip layout (e.g., see the titles and contents  ...  Introduction The problem of \nicely" drawing a graph in the plane has received increasing attention due to the large number of applications 3].  ... 
doi:10.1142/s0218195997000132 fatcat:jrhnpdtalnbepkhoivn2qfo6tu

Computing Radial Drawings on the Minimum Number of Circles

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer
2005 Journal of Graph Algorithms and Applications  
A radial drawing is a representation of a graph in which the vertices lie on concentric circles of finite radius.  ...  In this paper we study the problem of computing radial drawings of planar graphs by using the minimum number of concentric circles.  ...  We extend the block cut-vertex tree data structure [11] to identify specific subgraphs of G. Because of Lemma 1, we can (and will) restrict our attention to canonical graphs.  ... 
doi:10.7155/jgaa.00114 fatcat:4qsqknezznehhasrj2vvmi5nwq

Some Applications of Orderly Spanning Trees in Graph Drawing [chapter]

Ho-Lin Chen, Chien-Chih Liao, Hsueh-I Lu, Hsu-Chun Yen
2002 Lecture Notes in Computer Science  
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.  ...  Along this line of research, in this article we demonstrate two additional applications of orderly spanning trees in graph drawing, namely, Podevs drawing and rectangular dual.  ... 
doi:10.1007/3-540-36151-0_31 fatcat:biwe66jzfjgc3lydw6vongtdcu

An Application of Well-Orderly Trees in Graph Drawing [chapter]

Huaming Zhang, Xin He
2006 Lecture Notes in Computer Science  
Well-orderly trees seem to have the potential of becoming a powerful technique capable of deriving new results in graph encoding, graph enumeration and graph generation [3, 4] .  ...  In this paper, we reduce the height of the visibility representation of plane graphs from 5n/6 to (4n − 1)/5, by using well-orderly trees.  ...  In this paper, we are going to derive an application of well-orderly trees in graph drawing.  ... 
doi:10.1007/11618058_41 fatcat:v3iesk73afeybkzyt7qy6ktsi4

Exploring large graphs in 3D hyperbolic space

T. Munzner
1998 IEEE Computer Graphics and Applications  
In a graph a node can have many incoming links, but in a tree a canonical parent is chosen for each child. We call links that appear in the graph but not in the spanning tree nontree links.  ...  This "butterfly" configuration provides a canonical local orientation and also serves to minimize occlusion of both nodes and their text labels.  ... 
doi:10.1109/38.689657 fatcat:wxlf2i4x5rcajnbxsekl536qeq

Optimal Polygonal Representation of Planar Graphs [article]

Christian A. Duncan and Emden R. Gansner and Yifan Hu and Michael Kaufmann and Stephen G. Kobourov
2011 arXiv   pre-print
In this paper, we consider the problem of representing graphs by polygons whose sides touch.  ...  Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)xO(n) grid.  ...  Acknowledgements We would like to thank Therese Biedl for pointing out the very relevant work by Kant and Thomassen and the anonymous referees for their helpful and thoughtful comments.  ... 
arXiv:1104.1482v1 fatcat:smcicbqix5cgbetzxpzun35rdi

A Survey on Small-Area Planar Graph Drawing [article]

Giuseppe Di Battista, Fabrizio Frati
2014 arXiv   pre-print
We survey algorithms and bounds for constructing planar drawings of graphs in small area.  ...  In order to obtain a visibility representation R of G, Tamassia and Tollis design a very nice algorithm (an application is shown in Fig. 16 ).  ...  The result of Chrobak and Kant again relies on a canonical ordering.  ... 
arXiv:1410.1006v1 fatcat:pkvu3zq4ojdcfeo4nfhujda77q

Drawing series parallel digraphs symmetrically☆☆This research has been supported by an Australian Research Council Grant, KOSEF postdoctoral fellowship, and the SCARE project at the University of Limerick

Seok-Hee Hong, Peter Eades, Sang-Ho Lee
2000 Computational geometry  
We present several variations of the second step, with visibility drawings, "bus-orthogonal" drawings, and polyline drawings. All algorithms run in linear time.  ...  The next step uses these automorphisms to construct a symmetric drawing of the graph.  ...  . , v k be the children of r, in the order of their composition, and suppose that m = k/2 .  ... 
doi:10.1016/s0925-7721(00)00020-1 fatcat:7duutoj6pnd43obpgqwnznszrq
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