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Drawing Planar Cubic 3-Connected Graphs with Few Segments: Algorithms & Experiments Alex Igamberdiev Wouter Meulemans Andr e Schulz Graph complexity Complexity of a graph G = ( V, E ) Usually | V | , | E | , etc. Graph complexity

  1. Drawing Planar Cubic 3-Connected Graphs with Few Segments: Algorithms & Experiments Alex Igamberdiev Wouter Meulemans Andr´ e Schulz

  2. Graph complexity Complexity of a graph G = ( V, E ) Usually | V | , | E | , etc.

  3. Graph complexity Complexity of a graph G = ( V, E ) Usually | V | , | E | , etc. Says nothing about how complex a drawing is

  4. Visual complexity Planar graphs Number of geometric objects for drawing

  5. Visual complexity Planar graphs Number of geometric objects for drawing

  6. Visual complexity Planar graphs Number of geometric objects for drawing 1

  7. Visual complexity Planar graphs Number of geometric objects for drawing 2

  8. Visual complexity Planar graphs Number of geometric objects for drawing 3

  9. Visual complexity Planar graphs Number of geometric objects for drawing 4

  10. Visual complexity Planar graphs Number of geometric objects for drawing 5

  11. Visual complexity Planar graphs Number of geometric objects for drawing 6

  12. Visual complexity Planar graphs Number of geometric objects for drawing 7

  13. Visual complexity Planar graphs Number of geometric objects for drawing 8

  14. Visual complexity Planar graphs Number of geometric objects for drawing 9

  15. Visual complexity Planar graphs Number of geometric objects for drawing 9 line segments for 18 edges

  16. Known results Class Lower Upper Tree K/ 2 K/ 2 [Durocher et al, 2013] 2- and 3-trees 2 V 2 V [Dujmovi´ c et al, 2007] Segments 3-connected 2 V 5 V/ 2 [Dujmovi´ c et al, 2007] Triangulation 2 V 7 V/ 3 [Durocher, Mondal, 2014] Planar 2 V 16 V/ 3 − E [Durocher, Mondal, 2014]

  17. Known results Class Lower Upper Tree K/ 2 K/ 2 [Durocher et al, 2013] 2- and 3-trees 2 V 2 V [Dujmovi´ c et al, 2007] Segments 3-connected 2 V 5 V/ 2 [Dujmovi´ c et al, 2007] Triangulation 2 V 7 V/ 3 [Durocher, Mondal, 2014] Planar 2 V 16 V/ 3 − E [Durocher, Mondal, 2014] Circ. arcs 3-trees E/ 6 11 E/ 18 [Schulz, 2013] 3-connected E/ 6 2 E/ 3 [Schulz, 2013]

  18. Our results Line-segment drawings Planar cubic 3-connected graphs

  19. Our results Line-segment drawings Planar cubic 3-connected graphs Two new algorithms n/ 2 + 3 segments [Mondal et al, 2013] Resolve flaw & improved

  20. Our results Line-segment drawings Planar cubic 3-connected graphs Two new algorithms n/ 2 + 3 segments [Mondal et al, 2013] Resolve flaw & improved Experimental comparison

  21. Deconstruction algorithm

  22. Deconstruction algorithm Theorem. Every graph can be constructed from the triangular prism with insertions maintaining a given outer face. Insertion

  23. Deconstruction algorithm Algorithm 1. Draw triangular prism

  24. Deconstruction algorithm Algorithm 1. Draw triangular prism 2. Construct graph, maintaining drawing Inner faces are convex No insertions on outer face

  25. Deconstruction algorithm Algorithm 1. Draw triangular prism 2. Construct graph, maintaining drawing Insertion

  26. Deconstruction algorithm Algorithm 1. Draw triangular prism 2. Construct graph, maintaining drawing

  27. Deconstruction algorithm Algorithm 1. Draw triangular prism 2. Construct graph, maintaining drawing

  28. Deconstruction algorithm Algorithm 1. Draw triangular prism 2. Construct graph, maintaining drawing Insertion

  29. Deconstruction algorithm Algorithm 1. Draw triangular prism 2. Construct graph, maintaining drawing Insertion

  30. Windmill algorithm

  31. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  32. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  33. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  34. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  35. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  36. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  37. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  38. Windmill algorithm Algorithm cycle C drawn convex Pre: Post: inside of C drawn

  39. Postprocessing Set of harmonic equations [Aerts & Felsner, 2013 ] u = λv + (1 − λ ) w , for λ ∈ (0 , 1) w u v

  40. Postprocessing Set of harmonic equations [Aerts & Felsner, 2013 ] u = λv + (1 − λ ) w , for λ ∈ (0 , 1) Solve for uniform edge length, i.e. λ = 1 / 2 w w u u v v

  41. [Mondal et al, 2013]

  42. [Mondal et al, 2013] “Grid” n/ 2 + 4 segments 6 slopes ( n/ 2 + 1) 2 grid

  43. [Mondal et al, 2013] “Grid” n/ 2 + 4 segments 6 slopes ( n/ 2 + 1) 2 grid Resolved flaw in algorithm

  44. [Mondal et al, 2013] “Grid” “Min” n/ 2 + 4 segments n/ 2 + 3 segments 6 slopes 7 slopes ( n/ 2 + 1) 2 grid Not on a grid Resolved flaw in algorithm

  45. [Mondal et al, 2013] “Grid” “Min” n/ 2 + 4 segments n/ 2 + 3 segments 6 slopes 7 slopes ( n/ 2 + 1) 2 grid Not on a grid Resolved flaw in algorithm Reduced to 6 slopes On a grid

  46. Three algorithms Deconstruction Windmill [Mondal et al, 2013]

  47. Measuring layout quality 2000 graphs with 24 . . . 30 vertices using plantri Six measures for each graph-algorithm pair

  48. Measuring layout quality 2000 graphs with 24 . . . 30 vertices using plantri Six measures for each graph-algorithm pair Angular resolution

  49. Measuring layout quality 2000 graphs with 24 . . . 30 vertices using plantri Six measures for each graph-algorithm pair Angular resolution Edge length

  50. Measuring layout quality 2000 graphs with 24 . . . 30 vertices using plantri Six measures for each graph-algorithm pair Angular resolution Edge length Face aspect ratio

  51. Measuring layout quality 2000 graphs with 24 . . . 30 vertices using plantri Six measures for each graph-algorithm pair Angular resolution Edge length Face aspect ratio Average and worst-case

  52. Angular resolution Average WIN DEC DEC-ALT MON-GRID MON-MIN 0 π/ 2 Minimum WIN DEC DEC-ALT MON-GRID MON-MIN 0 π/ 2

  53. Edge length Average WIN DEC DEC-ALT MON-GRID MON-MIN 0 100% Maximum WIN DEC DEC-ALT MON-GRID MON-MIN 0 100%

  54. Face aspect ratio Average WIN DEC DEC-ALT MON-GRID MON-MIN 0 1 Minimum WIN DEC DEC-ALT MON-GRID MON-MIN 0 1

  55. Experiment summary WIN DEC MON WIN DEC MON WIN DEC MON “Wins” “Wins” minus “Losses” -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  56. Conclusion Minimal visual complexity Two new algorithms Fixed and improved [Mondal et al, 2013] Experiments Best depends on measure

  57. Conclusion Minimal visual complexity Two new algorithms Fixed and improved [Mondal et al, 2013] Experiments Best depends on measure Future work Closing gap for other classes Circular arcs Visual complexity ∼ observer’s assessment? Visual complexity ∼ cognitive load?

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