compact drawings of 1 planar graphs with right angle
play

Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and - PowerPoint PPT Presentation

Sep 21, 2023 •146 likes •1.55k views

Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends Steven Chaplick, Fabian Lipp, Alexander Wolff, and Johannes Zink Introduction: Beyond-Planar Graphs 2 Types of drawings: Planar: No crossings Introduction:

  1. Introduction: The Shift Algorithm 3 [de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea: v 4 v 6 • Triangulate given plane graph. • Compute a canonical ordering of v 3 v 5 the vertices v 1 , v 2 , . . . , v n . v 2 v 1 • Draw the graph: Resulting grid size: – Start with triangle v 1 , v 2 , v 3 . (2 n − 4) × ( n − 2) – For v k : v 6 Shift first & last neighbor of v k . – Add v k to the outer face. ⇒ all slopes on outer face ± 1 v 5 v 4 v 3 (except for v 1 v 2 ) v 1 v 2

  2. Introduction: The Shift Algorithm 3 [de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea: v 4 v 6 • Triangulate given plane graph. • Compute a canonical ordering of v 3 v 5 the vertices v 1 , v 2 , . . . , v n . v 2 v 1 • Draw the graph: Resulting grid size: – Start with triangle v 1 , v 2 , v 3 . (2 n − 4) × ( n − 2) – For v k : v 6 Shift first & last neighbor of v k . – Add v k to the outer face. ⇒ all slopes on outer face ± 1 v 5 v 4 v 3 (except for v 1 v 2 ) v 1 v 2

  3. Introduction: Related Work 4 1-planar = { graph G | G has a NIC-planar drawing } NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  4. Introduction: Related Work 4 RAC 2 RAC 1 1-planar RAC 0 NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  5. Introduction: Related Work 4 RAC poly 3 RAC 2 RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly 0 NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  6. Introduction: Related Work 4 RAC poly [de Fraysseix, Pach, Pollack, 3 1990] RAC 2 [Schnyder, 1990] RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly 0 NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  7. Introduction: Related Work 4 RAC poly RAC poly [Didimo, Eades, Liotta, = all graphs = all graphs E ? E ? 3 3 2009] RAC 2 RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly 0 NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  8. Introduction: Related Work 4 RAC poly [Didimo, Eades, Liotta, = all graphs E ? 3 2009] RAC 2 [Eades, Liotta, 2011] RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly 0 NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  9. Introduction: Related Work 4 RAC poly [Brandenburg, Didimo, = all graphs E ? 3 Evans, Kindermann, Liotta, RAC 2 Montecchiani, 2015] RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly 0 E ? E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  10. Introduction: Related Work 4 RAC poly [Liotta, Montecchiani, 2015] = all graphs E ? 3 RAC 2 RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly 0 E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  11. Introduction: Related Work 4 RAC poly [Didimo, Liotta, Mehrabi, = all graphs E ? 3 Montecchiani, 2016] RAC 2 RAC poly 2 RAC 1 E ? E ? RAC poly 1 1-planar RAC 0 RAC poly 0 E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  12. Introduction: Related Work 4 RAC poly [Bachmaier, Brandenburg, = all graphs E ? 3 Hanauer, Neuwirth, RAC 2 Reislhuber, 2017] RAC poly 2 RAC 1 E ? RAC poly 1 1-planar RAC 0 RAC poly 0 E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  13. Introduction: Related Work 4 RAC poly = all graphs E ? 3 RAC 2 RAC poly Our results 2 RAC 1 E ? RAC poly 1 1-planar RAC 0 RAC poly 0 E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  14. Introduction: Related Work 4 RAC poly = all graphs E ? 3 RAC 2 RAC poly Our results 2 RAC 1 E ? RAC poly 1 1-planar RAC 0 RAC poly w/o B-configuration 0 E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  15. Introduction: Related Work 4 RAC poly = all graphs E ? 3 RAC 2 RAC poly 2 RAC 1 E ? Our first main result: RAC poly 1 1-planar RAC 0 NIC-plane graphs RAC poly ⊆ RAC poly w/o B-configuration 0 1 E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  16. Introduction: Related Work 4 RAC poly Our second main result: = all graphs E ? 3 RAC 2 1-plane graphs RAC poly ⊆ RAC poly 2 2 RAC 1 E ? Our first main result: RAC poly 1 1-planar RAC 0 NIC-plane graphs RAC poly ⊆ RAC poly w/o B-configuration 0 1 E ? NIC-planar contained in (even for fixed embedding) IC-planar E ? contained in (unknown for fixed embedding) planar incomparable

  17. Result 1: NIC-Plane Graphs ⊆ RAC poly 5 1 RAC poly = all graphs E ? 3 RAC 2 RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly w/o B-configuration 0 NIC-planar IC-planar planar

  18. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph

  19. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works:

  20. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works:

  21. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite :

  22. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : v dummy

  23. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge v dummy

  24. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge v dummy

  25. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm v dummy

  26. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm v dummy

  27. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) v dummy

  28. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) v dummy

  29. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) v dummy

  30. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) grid point on the Thales’ circle v dummy

  31. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) grid point on the Thales’ circle v dummy

  32. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) grid points for the bends v dummy

  33. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy

  34. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy

  35. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy

  36. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � very slim v dummy

  37. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy

  38. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy

  39. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy

  40. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy

  41. Result 1: NIC-Plane Graphs ⊆ RAC poly 6 1 • Input: a NIC-plane graph Approach that nearly works: • Enclose each crossing by a so called empty kite : • Replace each pair of crossing edges by a single edge • Draw the obtained plane graph with the Shift Algorithm • Manually reinsert the removed edges with 1 bend so that they cross in a right angle (crossings and bends on the grid) � v dummy bad configu- ration!

  42. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad configu- ration!

  43. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices.

  44. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs).

  45. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

  46. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up start with instead of top-down. an empty quadrangle

  47. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down.

  48. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. Insert the diagonal

  49. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. Insert the diagonal

  50. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down.

  51. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down.

  52. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94

  53. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear:

  54. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1

  55. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1

  56. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1

  57. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1 Case 2

  58. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1 Case 2

  59. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1 Case 2

  60. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1 Case 2 Case 3

  61. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1 Case 2 Case 3

  62. Result 1: NIC-Plane Graphs ⊆ RAC poly 7 1 bad Solution: configu- ration! • Make the first vertex in the qudrangle (regarding the canonical ordering) adjacent to the other three vertices. v 55 v 61 • Use the algorithm by Harel and Sardas (Shift Algorithm for biconnected graphs). v 42 It builds a canonical ordering bottom-up instead of top-down. v 94 • Now only three “good” cases can appear: Case 1 Case 2 Case 3

  63. Result 1: NIC-Plane Graphs ⊆ RAC poly 8 1 Full example:

  64. Result 1: NIC-Plane Graphs ⊆ RAC poly 8 1 Full example:

  65. Result 2: 1-Plane Graphs ⊆ RAC poly 9 2 RAC poly = all graphs E ? 3 RAC 2 RAC poly 2 RAC 1 RAC poly 1 1-planar RAC 0 RAC poly w/o B-configuration 0 NIC-planar IC-planar planar

  66. Result 2: 1-Plane Graphs ⊆ RAC poly 10 2 • Input: a 1-plane graph

  67. Result 2: 1-Plane Graphs ⊆ RAC poly 10 2 • Input: a 1-plane graph Preprocessing:

  68. Result 2: 1-Plane Graphs ⊆ RAC poly 10 2 • Input: a 1-plane graph Preprocessing:

  69. Result 2: 1-Plane Graphs ⊆ RAC poly 10 2 • Input: a 1-plane graph Preprocessing: • Enclose each crossing by a so called subdivided kite :

Recommend

1-Bend RAC Drawings of NIC-Planar Graphs in Quadratic Area Steven Chaplick, Fabian Lipp,

1-Bend RAC Drawings of NIC-Planar Graphs in Quadratic Area Steven Chaplick, Fabian Lipp,

1-Bend RAC Drawings of NIC-Planar Graphs in Quadratic Area Steven Chaplick, Fabian Lipp, Alexander Wolff, and Johannes Zink Beyond-Planar Graphs 2 Types of Drawings: Planar: No crossings Beyond-Planar Graphs 2 Types of Drawings:

1.12k views • 107 slides
On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs Martin N

On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs Martin N

On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs Martin N ollenburg, Roman Prutkin , and Ignaz Rutter September 26, 2014 I NSTITUTE OF T HEORETICAL I NFORMATICS www.kit.edu KIT University of the State of

941 views • 69 slides
Non-aligned drawings of planar graphs Therese Biedl 1 , Claire Pennarun 2 1 University of Waterloo

Non-aligned drawings of planar graphs Therese Biedl 1 , Claire Pennarun 2 1 University of Waterloo

Non-aligned drawings of planar graphs Therese Biedl 1 , Claire Pennarun 2 1 University of Waterloo 2 LaBRI, Univ. Bordeaux Graph Drawing September 19, 2016 Claire Pennarun Non-aligned drawings of planar graphs September 19, 2016 1 / 11 N

734 views • 48 slides
1-Fan-Bundle-Planar Drawings of Graphs Patrizio Angelini Michael A. Bekos Michael Kaufmann

1-Fan-Bundle-Planar Drawings of Graphs Patrizio Angelini Michael A. Bekos Michael Kaufmann

1-Fan-Bundle-Planar Drawings of Graphs Patrizio Angelini Michael A. Bekos Michael Kaufmann Philipp Kindermann Thomas Schneck Beyond Planarity Planar Beyond Planarity 1-planar Beyond Planarity 2-planar Beyond Planarity k -planar Beyond

2.19k views • 162 slides
Smooth Orthogonal Drawings of Planar Graphs Philipp Kindermann Chair of Computer Science I

Smooth Orthogonal Drawings of Planar Graphs Philipp Kindermann Chair of Computer Science I

Smooth Orthogonal Drawings of Planar Graphs Philipp Kindermann Chair of Computer Science I Universit at W urzburg Joint work with Md. Jawaherul Alam, Michael A. Bekos, Michael Kaufmann, Stephen G. Kobourov & Alexander Wolff

1.48k views • 145 slides
Anchored Drawings of Planar Graphs Angelini, Da Lozzo, Di Bartolomeo, Di Battista, Hong,

Anchored Drawings of Planar Graphs Angelini, Da Lozzo, Di Bartolomeo, Di Battista, Hong,

22nd International Symposium on Graph Drawing 24-26 September 2014, Wrzburg, Germany Anchored Drawings of Planar Graphs Angelini, Da Lozzo, Di Bartolomeo, Di Battista, Hong, Patrignani, Roselli Applicative Context Drawing a graph on a

518 views • 48 slides
Planar Lombardi Drawings for Subcubic Graphs David Eppstein 20th International Symposium on Graph

Planar Lombardi Drawings for Subcubic Graphs David Eppstein 20th International Symposium on Graph

Planar Lombardi Drawings for Subcubic Graphs David Eppstein 20th International Symposium on Graph Drawing Redmond, Washington, September 1921, 2012 Mark Lombardi American neo-conceptual fine artist (19512000) Narrative structures,

375 views • 23 slides
Strongly Monotone Drawings of Planar Graphs Stefan Felsner Technische Universit at Berlin

Strongly Monotone Drawings of Planar Graphs Stefan Felsner Technische Universit at Berlin

Strongly Monotone Drawings of Planar Graphs Stefan Felsner Technische Universit at Berlin FernUniversit at in Hagen Alexander Igamberdiev Philipp Kindermann FernUniversit at in Hagen Freie Universit at Berlin Boris Klemz

732 views • 55 slides
Maximizing Ink in Partial Edge Drawings of k -plane Graphs Matthias Hummel, Fabian Klute, Soeren

Maximizing Ink in Partial Edge Drawings of k -plane Graphs Matthias Hummel, Fabian Klute, Soeren

Maximizing Ink in Partial Edge Drawings of k -plane Graphs Matthias Hummel, Fabian Klute, Soeren Nickel, Martin N ollenburg GD 2019 September 19, 2019 Partial Edge Drawings (PED) How to draw non-planar graphs? Just hide the edge crossings!

785 views • 77 slides
Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 1 , Sven Mallach 2 , Petra

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 1 , Sven Mallach 2 , Petra

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 1 , Sven Mallach 2 , Petra Mutzel 1 , Ulf uegg 3 , and Reinhard von Hanxleden 3 R 1 TU Dortmund University, Germany 2 University of Cologne, Germany 3 University of Kiel,

845 views • 18 slides
8.7 PLANAR GRAPHS def: A graph is planar if it can be drawn with- out edge-crossings in the

8.7 PLANAR GRAPHS def: A graph is planar if it can be drawn with- out edge-crossings in the

8.7.1 Section 8.7 Planar Graphs 8.7 PLANAR GRAPHS def: A graph is planar if it can be drawn with- out edge-crossings in the plane. Given a graph G and a Imbedding Problem: surface S , is it possible to draw G on S without any

372 views • 18 slides
Vertex Angle and Crossing Angle Resolution of Leveled Tree Drawings Walter Didimo U Perugia IT

Vertex Angle and Crossing Angle Resolution of Leveled Tree Drawings Walter Didimo U Perugia IT

Vertex Angle and Crossing Angle Resolution of Leveled Tree Drawings Walter Didimo U Perugia IT Michael Kaufmann U T ubingen DE Giuseppe Liotta U Perugia IT Yoshio Okamoto JAIST JP Andreas Spillner U Greifswald DE Didimo,

545 views • 36 slides
Crossing Numbers of Beyond-Planar Graphs Philipp Kindermann Universit at W urzburg joint

Crossing Numbers of Beyond-Planar Graphs Philipp Kindermann Universit at W urzburg joint

Crossing Numbers of Beyond-Planar Graphs Philipp Kindermann Universit at W urzburg joint work with Markus Chimani Fabrizio Montecchiani Pavel Valtr Crossing ratio Crossing number cr ( G ) : Min. # crossings over all drawings of G

928 views • 74 slides
LAGOS 2017 Delta-Wye Transformations and the Efficient Reduction of Almost-Planar Graphs Isidoro

LAGOS 2017 Delta-Wye Transformations and the Efficient Reduction of Almost-Planar Graphs Isidoro

Almost-planar graphs Deltawye reduction of almost-planar graph Almost-planar graphs on the projective plane Deltawye reduction with terminals of almost-planar graphs Future Research LAGOS 2017 Delta-Wye Transformations and the Efficient

1.3k views • 45 slides
1-bend Upward Planar Drawings of SP-digraphs with the Optimal Number of Slopes Emilio Di

1-bend Upward Planar Drawings of SP-digraphs with the Optimal Number of Slopes Emilio Di

1-bend Upward Planar Drawings of SP-digraphs with the Optimal Number of Slopes Emilio Di Giacomo, Giuseppe Liotta, Fabrizio Montecchiani Universit` a degli Studi di Perugia, Italy GD 2016, September 19-21, Athens How to draw a planar

767 views • 40 slides
1-bend RAC Drawings of 1-Planar Graphs Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, Fabrizio

1-bend RAC Drawings of 1-Planar Graphs Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, Fabrizio

1-bend RAC Drawings of 1-Planar Graphs Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, Fabrizio Montecchiani Things to avoid in graph drawing Things to avoid in graph drawing Too many crossings Things to avoid in graph drawing Too many

1.16k views • 66 slides
Level-Planar Drawings with Few Slopes Guido Br uckner Nadine Krisam Tamara Mchedlidze Level

Level-Planar Drawings with Few Slopes Guido Br uckner Nadine Krisam Tamara Mchedlidze Level

Level-Planar Drawings with Few Slopes Guido Br uckner Nadine Krisam Tamara Mchedlidze Level Graphs 4 directed graph G = ( V , E ) 3 level assignment : V N s.t. ( u , v ) E : ( u ) < ( v ) 2 1 Level

772 views • 37 slides
Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with with a Prescrib Prescribed ed

Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with with a Prescrib Prescribed ed

Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with with a Prescrib Prescribed ed Automorphism utomorphism Group Group C.J. Jones, Sarah E. Lubow, and Carlie J. Triplitt University of Texas at Tyler Automorphism Group of a

435 views • 27 slides
10 Steps to Counting Unlabeled Planar Graphs: 20 Years Later Manuel Bodirsky October 2007

10 Steps to Counting Unlabeled Planar Graphs: 20 Years Later Manuel Bodirsky October 2007

10 Steps to Counting Unlabeled Planar Graphs: 20 Years Later Manuel Bodirsky October 2007 Counting Unlabeled Planar Graphs A005470 Sloane Sequence A005470 (core, nice, hard): Number p ( n ) of unlabeled planar simple graphs with n nodes.

814 views • 70 slides
Topological Drawings of Complete Bipartite Graphs Jean Cardinal (ULB, Brussels) Joint work with

Topological Drawings of Complete Bipartite Graphs Jean Cardinal (ULB, Brussels) Joint work with

Topological Drawings of Complete Bipartite Graphs Jean Cardinal (ULB, Brussels) Joint work with Stefan Felsner (TU Berlin) June 18 1 / 26 Topological Drawings of Graphs vertices points edges (well-behaved) continuous curves 2 / 26

621 views • 26 slides
Geometric representations of planar graphs and maps Eric Fusy (CNRS/LIX) Summer school on

Geometric representations of planar graphs and maps Eric Fusy (CNRS/LIX) Summer school on

Geometric representations of planar graphs and maps Eric Fusy (CNRS/LIX) Summer school on random geometry, Bogota, may 2016 Overview of the course Planar graphs and planar maps - structural aspects 1 1 2 2 2 2 3 3 -

1.74k views • 160 slides
Chapter 7 Planar graphs In full: 7.17.3 Parts of: 7.4, 7.67.8 Skip: 7.5 Prof. Tesler

Chapter 7 Planar graphs In full: 7.17.3 Parts of: 7.4, 7.67.8 Skip: 7.5 Prof. Tesler

Chapter 7 Planar graphs In full: 7.17.3 Parts of: 7.4, 7.67.8 Skip: 7.5 Prof. Tesler Math 154 Winter 2020 Prof. Tesler Ch. 7: Planar Graphs Math 154 / Winter 2020 1 / 52 Planar graphs Definition A planar embedding of a graph is a

1.02k views • 52 slides
Cubic Planar Graphs That Cannot Be Drawn On Few Lines David Eppstein University of California,

Cubic Planar Graphs That Cannot Be Drawn On Few Lines David Eppstein University of California,

Cubic Planar Graphs That Cannot Be Drawn On Few Lines David Eppstein University of California, Irvine Symposium on Computational Geometry, June 2019 Planar graph drawing on one line Always possible using 2 semicircles per edge Planar graph

432 views • 16 slides
Densest/Heaviest k -subgraph on Interval Graphs, Chordal Graphs and Planar Graphs Presented by

Densest/Heaviest k -subgraph on Interval Graphs, Chordal Graphs and Planar Graphs Presented by

Densest/Heaviest k -subgraph on Interval Graphs, Chordal Graphs and Planar Graphs Presented by Jian Li, Fudan University Mar 2007, HKUST May 8, 2006 1 / 25 Problem Definition: Densest k -Subgraph Problem(DS- k ): Input: G ( V, E ) , k > 0 .

398 views • 25 slides

More recommend