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Department of Computer Science and Engineering Bangladesh University of Engineering and Technology

Point-Set Embeddings of Planar Graphs with Fewer Bends. M. Sc. Engg. Thesis Md. Emran Chowdhury (040805068P) Supervisor: Prof. Dr. Md. Saidur Rahman. Department of Computer Science and Engineering Bangladesh University of Engineering and Technology. Contents. Problem Definition.

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Department of Computer Science and Engineering Bangladesh University of Engineering and Technology

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  1. Point-Set Embeddings of Planar Graphs with Fewer Bends M. Sc. Engg. Thesis Md. Emran Chowdhury (040805068P) Supervisor: Prof. Dr. Md. Saidur Rahman Department of Computer Science and Engineering Bangladesh University of Engineering and Technology

  2. Contents • Problem Definition • Motivation • Previous Results and Our Results • Upward Point-Set Embedding • Orthogonal Point-Set Embedding • Conclusion and Future Works

  3. Point-Set Embedding Input d d f f e e a a G S c c b b Each vertex is placed at a distinct point

  4. Point-Set Embedding Input f d e f Bend e d c a G S b c b a Each vertex is placed at a distinct point Output Each edge is drawn by straight or poly line

  5. Upward Point-Set Embedding Input f d e f e d c a G S b c b a Each vertex is placed at a distinct point Output Each edge is drawn upward

  6. f d e f e d f c a e S G b c d b a c b a Upward Point-Set Embedding d f e Not every graph has upward point-set embedding on a fixed point-set a G S c f d e f b e d Each edge is drawn upward c a G’ has no upward point-set embedding on S S G’ b c b a

  7. Upward Point-Set Embedding with mapping 4 3 φ 2 b b S 1 a a c c G d d

  8. Upward Point-Set Embedding with mapping d Finding upward point-set embedding with mapping is a real challenge c φ b b S a a c G φ’ d b c No upward point-set embedding with this mapping d S a

  9. e f c d g h a b j k Orthogonal Point-Set Embedding Input planar graph G point-set in the plane

  10. j k e f h g c d g h f e a b j k d c a b Output Orthogonal Point-Set Embedding Each edge is drawn as a sequence of vertical and horizontal line segments

  11. Contents • Problem Definition • Motivation

  12. f f e e d d c b c b a a Motivation Point-Set Embedding In VLSI design, often the places for the modules are fixed, we have to connect the modules w. r. t. the inter connection graph Interconnection Graph

  13. f f e e d d c b c b a a Motivation Point-Set Embedding It is always desirable to reduce the number of bends Interconnection Graph VLSI Layout

  14. Contents • Problem Definition • Motivation • Previous Results and Our Results

  15. Previous Results and Our Results Upward Point-Set Embedding Authors Problem Graph class Results upward point-set embedding Upward planar digraphs at most two bends per edge Giordano et. al. ’07 upward point-set embedding with mapping upper bound on total number of bends Upward planar digraphs at most 2n-3 bends per edge Giordano, Liotta, and Whiteside ’09 upward point-set embedding with mapping Upward planar digraphs This Thesis at most n-3 bends per edge

  16. Previous Results and Our Results Orthogonal Drawing Authors Problem Graph class Results Orthogonal drawing Cubic 3-connected plane graphs bend optimal drawing Rahman et. al. ’99 Time complexity = O(n) But, they did not consider the point-set embedding Orthogonal drawing plane graphs with  ≤ 3 bend optimal drawing Rahman and Nishizeki ’02 Orthogonal drawing plane graphs with  ≤ 3 no bend drawing Rahman, Nishizeki and Naznin ’03

  17. Previous Results and Our Results Poly-line Point-Set Embedding Authors Problem Graph class Results Point-set embedding General plane graphs 2 bends per edge Kaufman and Wiese ’02 But, the size of the vertices may increase One can draw the edge orthogonally Time complexity = O(n2)

  18. Previous Results and Our Results Orthogonal Point-Set Embedding Authors Problem Graph class Results Point-set embedding General plane graphs 2 bends per edge Kaufman and Wiese ’02 But, the size of the vertices may increase One can draw the edge orthogonally Time complexity = O(n2) This Thesis Orthogonal point-set embedding 3-connected cubic planar graphs at most (5n+4)/2 bends in total Time complexity = O(n) Tight upper bound This Thesis Orthogonal point-set embedding with mapping 4-connected planar graphs at most 6n bends in total

  19. Contents • Problem Definition • Motivation • Previous Results and Our Results • Upward Point-Set Embedding

  20. 5 4 3 2 1 5 5 4 4 3 v3 v3 v5 v5 3 2 1 v1 v1 v2 v2 2 v4 v4 1 Upward Point-Set Embedding G Input S Upward Topological Book Embedding Upward Point-set Embedding

  21. Upward Topological Book Embedding b d a c G c Left Page Right Page d S Digraph b The vertices on the spine a The edges on the pages Upward Topological Book Embedding Spine

  22. Upward Topological Book Embedding G contains directed hamiltonian path G contains directed hamiltonian path 7 A directed path containing all the vertices 7 6 6 5 5 4 4 3 3 2 1 2 1

  23. Upward Topological Book Embedding G contains directed hamiltonian path G contains directed hamiltonian path 7 7 6 6 5 5 4 4 3 3 2 1 2 1

  24. 7 6 5 4 3 2 1 Upward Topological Book Embedding

  25. 7 6 5 4 3 2 1 Upward Topological Book Embedding 7 6 5 4 3 2 1

  26. The drawing ….. • has no edge crossings since • it has the same embedding • as the original graph • has no spine crossing • has 1 bend per edge 7 6 5 4 3 2 1 Upward Topological Book Embedding 7 6 5 4 3 2 1

  27. Upward Topological Book Embedding G does not contain directed Hamiltonian path 7 6 5 e 4 b d c 3 2 a 1

  28. Upward Topological Book Embedding G does not contain directed Hamiltonian path 7 6 5 e 4 b d c 3 2 a 1

  29. Upward Topological Book Embedding G does not contain directed Hamiltonian path 7 6 5 e 4 b d c 3 2 a 1

  30. Upward Topological Book Embedding 7 7 6 6 5 e 5 e 4 4 d b d c c 3 b 2 a 3 1 a 2 1

  31. 7 7 7 6 6 6 5 5 e 5 e 4 4 4 d b d c c 3 b 2 a 3 3 1 a 2 2 1 1 Upward Topological Book Embedding Input digraph Each spine crossing corresponds to a dummy vertex

  32. Calculation of number of Bends j Spine crossing from i to j is at most j-i-2 j-1 The edge (1, n) has no crossings j-2 Spine Crossings per edge is at most n-4 i+2 i+1 i

  33. Calculation of number of Bends Spine crossing from i to j is at most j-i-2 The edge (1, n) has no crossings Spine Crossings per edge is at most n-4 Bends per edge is at most n-3

  34. Calculation of number of Bends Spine crossing from i to j is at most j-i-2 Total number of spine crossings =2(n-4)+3(n-5)+ . . . +k(n-2-k)+p(n-3-k) where p, k are integers Number of edges which crosses the spine={k(k+1)/2}-1+p The edge (1, n) has no crossings Spine Crossings per edge is at most n-4 Bends per edge is at most n-3

  35. Contents • Problem Definition • Motivation • Previous Results and Our Results • Upward Point-Set Embedding • Orthogonal Point-Set Embedding

  36. Orthogonal Point-Set Embedding 3-connected cubic planar graphs 4-connected planar graphs ( ≤ 4) 4-connected 4-regular planar graphs

  37. v5 v6 v3 v4 v7 v8 v2 v1 v10 v9 3-connected cubic planar graph 3-connected cubic planar graph G with HC

  38. v5 v6 v3 v4 v7 v8 v2 v1 v10 v9 3-connected cubic planar graph 3-connected cubic planar graph G with HC Plane embedding G’ of graph G point-set in the plane

  39. 3-connected cubic planar graph v5 v6 Inner edges v3 v4 v7 v8 v2 v1 v10 v9 Outer edges Plane embedding G’ of graph G

  40. 3-connected cubic planar graph v5 v6 v3 v4 v7 v8 v2 v1 v10 v9 Inner vertices Plane embedding G’ of graph G

  41. 3-connected cubic planar graph p10 p9 p8 v5 v6 p7 v3 v4 v7 v8 p6 p5 v2 v1 v10 v9 p4 p3 p2 p1

  42. 3-connected cubic planar graph p10 p9 p8 v5 v6 We have to consider two cases Case 1: Inner edges in left page Case 2 : Inner edges in right page p7 v3 v4 v7 v8 p6 p5 v2 v1 v10 v9 p4 p3 p2 p1 Case 1: Inner edges in left page

  43. 3-connected cubic planar graph p10 Nice points (L) p9 p8 v5 v6 p7 v3 v4 v7 v8 p6 p5 v2 v1 v10 v9 p4 p3 CountL= 6 p2 p1 Case 1: Inner edges in left page

  44. 3-connected cubic planar graph p10 p9 p8 v5 v6 p7 v3 v4 v7 v8 p6 Nice points (R) p5 v2 v1 v10 v9 p4 p3 CountL= 6 p2 CountR= 2 p1 Case 2: Inner edges in right page

  45. p10 p9 p8 v5 v6 p7 v3 v4 v7 v8 p6 p5 v2 v1 v10 v9 p4 p3 p2 p1 3-connected cubic planar graph CountL= 6 CountR= 2

  46. 3-connected cubic planar graph p10 Computation of number of bends p9 • From pigeonhole principle….. • Either countL or • countR is at • least = (n-2)/2 • which edges can • be drawn with 1 bend p8 v5 v6 Left nice points Total bends = 1.(n-2)/2+2.(3n/2-(n-2)/2-1)+3 = n/2-1+3n-n+2-2+3 = n/2+2n+2 = (5n+4)/2 p7 v3 v4 v7 v8 p6 p5 v2 v1 v10 v9 Right nice points p4 p3 p2 p1

  47. 6 5 4 3 2 1 4-connected planar graph v2 v4 v3 v6 v1 v5 4-connected planar graph G Point-set S

  48. 6 5 4 3 2 1 4-connected planar graph v2 v4 v3 v6 v1 v5 Plane embedding G’ of graph G Point-set S

  49. 6 v6 v2 5 v4 v5 v3 4 v4 v6 3 v1 v5 v3 2 v2 1 v1 4-connected planar graph Outer edges Inner edges Plane embedding G’ of graph G

  50. v6 v5 v4 v3 v2 v1 4-connected planar graph Outer edges v2 v4 v3 Inner edges v6 v1 v5

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