1. D.I.E.I. - Università degli Studi di Perugia h-quasi planar drawings of bounded treewidth graphs in linear area Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani University of Perugia 13 th Italian Conference on Theoretical Computer Science 19-21 September 2012, Varese, Italy

2. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Graph Drawing and Area Requirement 19/09/20122 Graph G Straight-line grid drawing of G

3. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Graph Drawing and Area Requirement Area requirement of straight-line drawings is a widely studied topic in Graph Drawing 19/09/20123 Graph G Straight-line grid drawing of G h w

4. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area Requirement for planar drawings Area requirement problem mainly studied for planar straight- line grid drawings: – planar graphs have planar straight-line grid drawings in O(n 2 ) area (worst case optimal) [de Fraysseix et al.; Schnyder; 1990] – sub-quadratic upper bounds: trees – O(n log n) [Crescenzi et al., 1992] outerplanar graphs – O(n 1.48 ) [Di Battista, Frati, 2009] – super-linear lower bound: series-parallel graphs – Ω(n2 (log n) ) [Frati, 2010] 19/09/20124

5. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area Requirement for planar drawings Planarity imposes severe limitations on the optimization of the area 19/09/20125

6. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area Requirement for planar drawings Planarity imposes severe limitations on the optimization of the area – Non-planar straight-line drawings in O(n) area exist for k-colorable graphs [Wood, 2005] – no guarantee on the type and on the number of crossings 19/09/20126 A drawing by Woods technique

7. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Beyond planarity: crossing complexity Non-planar drawings should be considered: – How can we control the crossing complexity of a drawing? 19/09/20127

8. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Crossing complexity measures Large Angle Crossing drawings (LAC) or Right Angle Crossing drawings (RAC), [Didimo et al., 2011] 19/09/20128 RAC drawing

9. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Crossing complexity measures h-Planar drawings: at most h crossings per edge 19/09/20129 1-planar drawing

10. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Crossing complexity measures h-Quasi Planar drawings: at most h-1 mutually crossing edges 19/09/201210 3-quasi planar drawing

11. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy The problem We investigate trade-offs between area requirement and crossing complexity We focus on h-quasi planarity as a measure of crossing complexity 19/09/201211

12. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Our contribution 1/2 (h-quasi planar drawings) General technique: Every n-vertex graph with treewidth k, has an h-quasi planar drawing in O(n) area with h depending only on k 19/09/201212

13. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Our contribution 1/2 (h-quasi planar drawings) General technique: Every n-vertex graph with treewidth k, has an h-quasi planar drawing in O(n) area with h depending only on k Ad-hoc techniques: Smaller values of h for specific subfamilies of planar partial k-trees (outerplanar, flat series-parallel, proper simply nested) 19/09/201213

14. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Our contribution 2/2 (h-quasi planarity vs h-planarity) Comparison: There exist n-vertex series-parallel graphs (partial 2-trees) such that every h-planar drawing requires super-linear area for any constant h – 11-quasi planar drawings in linear area always exist 19/09/201214

15. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Our contribution 2/2 (h-quasi planarity vs h-planarity) Comparison: There exist n-vertex series-parallel graphs (partial 2-trees) such that every h-planar drawing requires super-linear area for any constant h – 11-quasi planar drawings in linear area always exist Additional result: There exist n-vertex planar graphs such that every h-planar drawing requires quadratic area for any constant h 19/09/201215

16. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Whats coming next Basic definitions Results on h-quasi planarity Comparison with h-planarity Conclusions and open problems 19/09/201216

17. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy BASIC DEFINITIONS 19/09/201217

18. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Bounded treewidth graphs 19/09/201218 Whats a k-tree?

19. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Bounded treewidth graphs 19/09/201219 Whats a k-tree? a clique of size k is a k-tree 3-tree construction

20. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Bounded treewidth graphs 19/09/201220 Whats a k-tree? a clique of size k is a k-tree the graph obtained from a k-tree by adding a new vertex adjacent to each vertex of a clique of size k is a k-tree 3-tree construction

21. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Bounded treewidth graphs 19/09/201221 Whats a k-tree? a clique of size k is a k-tree the graph obtained from a k-tree by adding a new vertex adjacent to each vertex of a clique of size k is a k-tree 3-tree construction

22. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Bounded treewidth graphs 19/09/201222 Whats a k-tree? a clique of size k is a k-tree the graph obtained from a k-tree by adding a new vertex adjacent to each vertex of a clique of size k is a k-tree 3-tree construction

23. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Bounded treewidth graphs 19/09/201223 Whats a k-tree? a clique of size k is a k-tree the graph obtained from a k-tree by adding a new vertex adjacent to each vertex of a clique of size k is a k-tree 3-tree construction

24. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Bounded treewidth graphs 19/09/201224 Whats a k-tree? a clique of size k is a k-tree the graph obtained from a k-tree by adding a new vertex adjacent to each vertex of a clique of size k is a k-tree A subgraph of a k-tree is a partial k-tree A graph has treewidth k it is a partial k-tree 3-tree construction

25. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Track assignment t-track assignment of a graph G [Dujmović et al., 2004] = t vertex coloring + total ordering < i in each color class V i 19/09/201225 3-track assignment

26. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Track assignment t-track assignment of a graph G [Dujmović et al., 2004] = t vertex coloring + total ordering < i in each color class V i – (V i, < i ) = track τ i, 1 i t 19/09/201226 3-track assignment track

27. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Track assignment t-track assignment of a graph G [Dujmović et al., 2004] = t vertex coloring + total ordering < i in each color class V i – (V i, < i ) = track τ i, 1 i t – X-crossing = (u, v), (w, z): u,w V i, v, z V j, u < i w and z < j v, for i j 19/09/201227 X-crossing

28. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Track assignment t-track assignment of a graph G [Dujmović et al., 2004] = t vertex coloring + total ordering < i in each color class V i – (V i, < i ) = track τ i, 1 i t – X-crossing = (u, v), (w, z): u,w V i, v, z V j, u < i w and z < j v, for i j 19/09/201228 NOT an X-crossing

29. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Track layout (c, t)-track layout of G = t-track assignment + edge c-coloring: no two edges of the same color form an X-crossing 19/09/201229 (2,3)-track layout

30. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Track layout (c, t)-track layout of G = t-track assignment + edge c-coloring: no two edges of the same color form an X-crossing 19/09/201230 (2,3)-track layout

31. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Track layout (c, t)-track layout of G = t-track assignment + edge c-coloring: no two edges of the same color form an X-crossing 19/09/201231 (2,3)-track layout

32. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy THE GENERAL TECHNIQUE: COMPUTING COMPACT H-QUASI PLANAR DRAWINGS OF K-TREES 19/09/201232

33. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Ingredients of the result 19/09/201233 assume to have a (c,t)-track layout: we show how to compute a [c(t-1)+1]-quasi planar drawing in O(t 3 n) area

34. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Ingredients of the result 19/09/201234 assume to have a (c,t)-track layout: we show how to compute a [c(t-1)+1]-quasi planar drawing in O(t 3 n) area we prove that every partial k-tree has a (2,t)-track layout where t depends on k but it does not depend on n

35. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Ingredients of the result 19/09/201235 assume to have a (c,t)-track layout: we show how to compute a [c(t-1)+1]-quasi planar drawing in O(t 3 n) area we prove that every partial k-tree has a (2,t)-track layout where t depends on k but it does not depend on n every partial k-tree has a O(1)-quasi planar drawing in area O(n)

36. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201236 INPUT: A partial k-tree G

37. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201237 G = 2-tree INPUT: A partial k-tree G

38. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201238 INPUT: A partial k-tree G 1.Compute a (2,t k )-track layout of G

39. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201239 1) = (2,t)-track layout of G t = 4

40. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201240 INPUT: A partial k-tree G 1.Compute a (2,t k )-track layout of G 2.Construct an h k -quasi planar drawing from OUTPUT: The drawing

41. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201241 2) = h-quasi planar drawing of G h c(t-1)+1 = 2(4-1)+1 = 7

42. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201242 INPUT: A partial k-tree G 1.Compute a (2,t k )-track layout of G 2.Construct an h k -quasi planar drawing from OUTPUT: The drawing

43. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing 19/09/201243 Lemma 1: every n-vertex graph G admitting a (c,t)-track layout, also admits an h-quasi planar drawing in O(t 3 n) area, where h = c(t 1) + 1

44. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy An example 19/09/201244

45. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing 19/09/201245 place the vertices along segments

46. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing 19/09/201246 any edge connecting a vertex on a segment i to a vertex on a segment j (i < j) do not overlap with any vertex on a segment k s.t. i < k <j

47. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing 19/09/201247 any edge connecting a vertex on a segment i to a vertex on a segment j (i < j) do not overlap with any vertex on a segment k s.t. i < k <j

48. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing 19/09/201248

49. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing 19/09/201249 O(t2n)O(t2n) t A = O(t 3 n)

50. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h We prove that at most c(t 1) edges mutually cross 19/09/201250

51. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h We prove that at most c(t 1) edges mutually cross – every edge (u,v) with u ϵ s i and v ϵ s j is completely contained in a parallelogram Π i,j 19/09/201251 sisi parallelogram Π i,j sjsj

52. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h 19/09/201252 at most c mutually crossing edges in each parallelogram

53. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h 19/09/201253 at most c mutually crossing edges in each parallelogram + at most t 1 parallelograms mutually overlap (to prove)

54. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h 19/09/201254 at most c mutually crossing edges in each parallelogram + at most t 1 parallelograms mutually overlap (to prove) at most c(t 1) mutually crossing edges in our drawing =

55. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h Simplified (but consistent) model – segments = points 19/09/201255

56. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h Simplified (but consistent) model – segments = points – parallelograms = curves 19/09/201256

57. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h An overlap occurs iff 1 - two curves form a crossing 19/09/201257

58. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h An overlap occurs iff 2 - two curves share an endpoint and the other two endpoints are either before or after the one in common 19/09/201258

59. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h Simplified (but consistent) model – an overlap occurs iff 1 - two curves form a crossing 2 - two curves share an endpoint and the other two endpoints are either before or after the one in common 19/09/201259 4 mutually overlapping parallelograms

60. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h To prove: at most t 1 parallelograms mutually overlap Proof by induction on t 19/09/201260

61. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h To prove: at most t 1 parallelograms mutually overlap Proof by induction on t – t = 2: one parallelogram, OK 19/09/201261

62. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h To prove: at most t 1 parallelograms mutually overlap Proof by induction on t – t = 2: one parallelogram, OK – t > 2: O t = biggest set of mutually overlapping parallelograms in Γ t – suppose by contradiction that |O t | > t – 1 By induction |O t-1 | t - 2 19/09/201262

63. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h 19/09/201263 12 i1i1 i2i2 ipip i p + 1 t-1 t O t = P U R

64. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h 19/09/201264 P = subset of parallelograms of O t having s t as a side – t 2+ |P| t |P| 2 12 i1i1 i2i2 ipip i p + 1 t-1 t

65. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h 19/09/201265 P = subset of parallelograms of O t having s t as a side – t 2+ |P| t |P| 2 R = O t \ P – they must have a side s j, 1 j i 1 and a side s l, i p + 1 l t 1 they are present in Γ t-1 – |O t | = |R| + |P| and |O t | t |R| t |P| 12 i1i1 i2i2 ipip i p + 1 t-1 t

66. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (c,t)-track layout h-quasi planar drawing : upper bound on h 19/09/201266 Let i h + 1 l t 1 be the greatest index among the segments in R – parallelograms Π i 2,l,…, Π i p,l and all the parallelograms in R mutually overlap they form a bundle of mutually overlapping parallelograms in Γ t1 whose size is at least t |P| + |P| 1 > t - 2, a contradiction, OK 12 i1i1 i2i2 ipip i p + 1 t-1 t

67. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy (2, t k )-track layout of k-trees Theorem 1: Every partial k-tree admits a (2, t k )-track layout, where t k is given by the following set of equations: 19/09/201267

68. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Putting results together Theorem 2: Every partial k-tree with n vertices admits a h k -quasi planar grid drawing in O(t k 3 n) area, where h k = 2(t k 1) + 1 and t k is given by the following set of equations: 19/09/201268

69. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Some values 19/09/201269 Kh_k (our result)h_k [Di Giacomo et al., 2005] 133 21115 32995415 (1,t)-track layouts

70. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy COMPARING H-QUASI PLANARITY WITH H-PLANARITY 19/09/201270

71. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees Theorem 6: Let h be a positive integer, there exist n- vertex series-parallel graphs such that any h-planar straight-line drawing requires Ω(n2 (log n) ) area Hence, h-planarity is more restrictive than h-quasi planarity in terms of area requirement for partial 2-trees 19/09/201271

72. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof 19/09/201272 a graph G

73. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof 19/09/201273 l …. G* = l-extension of G

74. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h- extension G * of G, there are no two edges of G crossing each other. 19/09/201274

75. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h- extension G * of G, there are no two edges of G crossing each other. 19/09/201275 if 2 edges of G cross… u v w z

76. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h- extension G * of G, there are no two edges of G crossing each other 19/09/201276 …one vertex will be inside a triangle u v w z

77. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h- extension G * of G, there are no two edges of G crossing each other 19/09/201277 …at least one edge of the triangle will receive h+1 crossings…!!! h u v w z

78. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof Consider the n-vertex graph G of the family of series- parallel graphs described in [Frati, 2010] – Ω(n2 (log n) ) area may be required in planar s.l. drawings 19/09/201278 G

79. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Area lower bound for h-planar drawings of partial 2-trees: sketch of proof Construct the 3h-extension G * of G – n * = 3m + n = Θ(n) – G * is a series-parallel graph – G must be drawn planarly in any h-planar drawing of G * 19/09/201279 G 3h ….

80. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Extending the lower bound to planar graphs Theorem 7: Let ε > 0 be given and let h(n) : N N be a function such that h(n) n 0.5 ε n ϵ N. For every n > 0 there exists a graph G with Θ(n) vertices such that any h(n)-planar straight-line grid drawing of G requires Ω(n 1+ 2ε ) area – Ω(n 2 ) area necessary if h is a constant 19/09/201280 3h ….

81. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy CONCLUSIONS AND OPEN PROBLEMS 19/09/201281

82. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Conclusions and remarks We studied h-quasi planar drawings of partial k-trees in linear area – drawings with optimal area and controlled crossing complexity 19/09/201282

83. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Conclusions and remarks We studied h-quasi planar drawings of partial k-trees in linear area – drawings with optimal area and controlled crossing complexity Interesting also in the case of planar graphs – Are there h-quasi planar drawings of planar graphs in o(n 2 ) area where h ϵ o(n)? 19/09/201283

84. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Conclusions and remarks We studied h-quasi planar drawings of partial k-trees in linear area – drawings with optimal area and controlled crossing complexity Interesting also in the case of planar graphs – Are there h-quasi planar drawings of planar graphs in o(n 2 ) area where h ϵ o(n)? O(n) area and h ϵ O(1) can be simultaneously achieved for some families of planar graphs 19/09/201284

85. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Conclusions and remarks We studied h-quasi planar drawings of partial k-trees in linear area – drawings with optimal area and controlled crossing complexity Interesting also in the case of planar graphs – Are there h-quasi planar drawings of planar graphs in o(n 2 ) area where h ϵ o(n)? O(n) area and h ϵ O(1) can be simultaneously achieved for some families of planar graphs Theorem 8: Every planar graph with n vertices admits a O(log 16 n)-quasi planar grid drawing in O(n log 48 n) area 19/09/201285

86. E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia ICTCS 12 - Varese, Italy Some open problems h-quasi planar drawings of planar graphs: – is it possible to achieve both O(n) area and h ϵ O(1)? h-quasi planar drawings of partial k-trees: – studying area - aspect ratio trade offs: O(n) area and o(n) aspect ratio? 19/09/201286