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

Transcription

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

2 Things to avoid in graph drawing

3 Things to avoid in graph drawing Too many crossings

4 Things to avoid in graph drawing Too many crossings Too many bends

5 A good property

6 A good property Right angle crossings (RAC)! [Huang, Hong, Eades 2008]

7 A good property Right angle crossings (RAC)! 1-bend 1-planar RAC drawing [Huang, Hong, Eades 2008]

8 Questions General: Which kind of graphs can be drawn with: a few crossings per edge, a few bends per edge, right angle crossings (RAC)?

9 Questions General: Which kind of graphs can be drawn with: a few crossings per edge, a few bends per edge, right angle crossings (RAC)? Specific: Does every 1-planar graph admit a 1-bend RAC drawing?

10 1-planar RAC drawings Not all 1-planar graphs have a straight-line RAC drawing [consequence of edge density results] Not all straight-line RAC drawable graphs are 1-planar [Eades and Liotta ] Every 1-plane kite-triangulation has a 1-bend RAC drawing [Angelini et al ] Every 1-plane graph with independent crossings (IC-planar) has a straight-line RAC drawing [Brandenburg et al ]

11 Our result Theorem. Every simple 1-planar graph admits a 1-bend RAC drawing, which can be computed in linear time if an initial 1-planar embedding is given

12 Some definitions 1-plane graph (not necessarily simple)

13 Some definitions 1-plane graph (not necessarily simple) kite

14 Some definitions empty kite 1-plane graph (not necessarily simple)

15 Some definitions 1-plane graph (not necessarily simple) not a kite!

16 Observation triangulated 1-plane graph (not necessarily simple)

17 Observation triangulated 1-plane graph (not necessarily simple) empty kite Every pair of crossing edges forms an empty kyte except possibly for a pair of crossing edges on the outer face

18 Observation triangulated 1-plane graph (not necessarily simple) not a kite Every pair of crossing edges forms an empty kyte except possibly for a pair of crossing edges on the outer face

19 Algorithm Outline input G simple 1-plane augmentation (the embedding may change) 1 G + triangulated 1-plane (multi-edges) recursive procedure 2 1-bend 1-planar RAC drawing of G 4 removal of dummy elements + 1-bend 1-planar RAC drawing of G + recursive procedure G * hierarchical contraction of G + 3 output

20 Algorithm Outline input G simple 1-plane augmentation (the embedding may change) 1 G + triangulated 1-plane (multi-edges) recursive procedure 2 1-bend 1-planar RAC drawing of G 4 removal of dummy elements + 1-bend 1-planar RAC drawing of G + recursive procedure G * hierarchical contraction of G + 3 output

21 G simple 1-plane Augmentation

22 Augmentation G simple 1-plane for each pair of crossing edges add an enclosing 4-cycle

23 Augmentation G simple 1-plane for each pair of crossing edges add an enclosing 4-cycle

24 Augmentation G simple 1-plane for each pair of crossing edges add an enclosing 4-cycle

25 Augmentation G simple 1-plane for each pair of crossing edges add an enclosing 4-cycle

26 G simple 1-plane Augmentation remove those multiple edges that belong to the input graph

27 G simple 1-plane Augmentation

28 G simple 1-plane Augmentation remove one (multiple) edge from each face of degree two, if any

29 Augmentation G simple 1-plane triangulate faces of degree > 3 by inserting a star inside them

30 G + triangulated 1-plane Augmentation

31 Algorithm Outline input G simple 1-plane augmentation (the embedding may change) 1 G + triangulated 1-plane (multi-edges) recursive procedure 2 1-bend 1-planar RAC drawing of G 4 removal of dummy elements + 1-bend 1-planar RAC drawing of G + recursive procedure G * hierarchical contraction of G + 3 output

32 Property of G + G + triangulated 1-plane - triangular faces - multiple edges never crossed - only empty kites

33 Property of G + G + triangulated 1-plane - triangular faces - multiple edges never crossed - only empty kites structure of each separation pair

34 Property of G + G + triangulated 1-plane - triangular faces - multiple edges never crossed - only empty kites structure of each separation pair

35 G + triangulated 1-plane Hierarchical contraction contract all inner components of each separation pair into a thick edge structure of each separation pair

36 G + triangulated 1-plane Hierarchical contraction contract all inner components of each separation pair into a thick edge contraction

37 G + triangulated 1-plane Hierarchical contraction contract all inner components of each separation pair into a thick edge contraction

38 G + triangulated 1-plane Hierarchical contraction

39 G + triangulated 1-plane Hierarchical contraction

40 Hierarchical contraction G + triangulated 1-plane G * hierarchical contraction

41 Hierarchical contraction G + triangulated 1-plane G * hierarchical contraction simple 3-connected triangulated 1-plane graph

42 Algorithm Outline input G simple 1-plane augmentation (the embedding may change) 1 G + triangulated 1-plane (multi-edges) recursive procedure 2 1-bend 1-planar RAC drawing of G 4 removal of dummy elements + 1-bend 1-planar RAC drawing of G + recursive procedure G * hierarchical contraction of G + 3 output

43 Drawing procedure remove crossing edges 3-connected plane graph apply Chiba et al convex faces and prescribed outerface reinsert crossing edges partial drawing

44 partial drawing Drawing procedure

45 partial drawing Drawing procedure

46 partial drawing Drawing procedure

47 Drawing procedure partial drawing remove crossing edges

48 partial drawing Drawing procedure

49 Drawing procedure partial drawing apply Chiba et al. 1984

50 Drawing procedure partial drawing reinsert crossing edges

51 partial drawing Drawing procedure

52 partial drawing Drawing procedure

53 Drawing procedure partial drawing remove crossing edges

54 partial drawing Drawing procedure

55 Drawing procedure partial drawing apply Chiba et al. 1984

56 Drawing procedure partial drawing reinsert crossing edges

57 new partial drawing Drawing procedure

58 new partial drawing Drawing procedure

59 Drawing procedure new partial drawing draw it as usual

60 + 1-bend 1-planar RAC drawing of G + Drawing procedure

61 Algorithm Outline input G simple 1-plane augmentation (the embedding may change) 1 G + triangulated 1-plane (multi-edges) recursive procedure 2 1-bend 1-planar RAC drawing of G 4 removal of dummy elements + 1-bend 1-planar RAC drawing of G + recursive procedure G * hierarchical contraction of G + 3 output

62 Drawing procedure + 1-bend 1-planar RAC drawing of G + remove dummy elements

63 Drawing procedure 1-bend 1-planar RAC drawing of G input graph G

64 input graph G Drawing procedure

65 Open problems Our algorithm may give rise to drawings with exponential area: is such an area necessary in some cases? Our algorithm is allowed to change the initial embedding: What if we cannot? Still missing: Characterization of straight-line 1-planar RAC graphs

66 Thank you