On Upward Point-Set Embedding of Upward Planar Digraphs

1. On Upward Point-Set Embedding of Upward Planar Digraphs Md. EmranChowdhury Muhammad JawaherulAlam Md. SaidurRahman Department of Computer Science & Engineering Bangladesh University of Engineering & Technology (BUET)

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

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

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

5. f f e e d d c c S S b b a a Upward Point-Set Embedding d f e There is an Upward Point-set Embedding of G on S if and only if G is upward planar There is an Upward Point-set Embedding of G on S if and only if G is acyclic a G c b

6. Upward Point-Set Embedding f d e f e d f c a e G S b c d b a c S b a

7. Upward Point-Set Embedding Giordano et. al. Upward Point-Set Embedding of any upward planar digraph with on any point set with at most two bends per edge f d e f e d c a G S b c b a

8. Upward Point-Set Embedding with mapping d c φ b b b S a a a c c G d d

9. Upward Point-Set Embedding with mapping d c φ b b S a a c G φ’ d b c No upward point-set embedding with this mapping d S a

10. Upward Point-Set Embedding with mapping • Giordano et. al. • O(n3)-time testing algorithm • O(n2)-time drawing algorithm • (2n-3) bends per edge Upward Topological Book Embedding with a given ordering • Ours • O(n2)-time drawing algorithm • (n-3) bends per edge

11. Upward Topological Book Embedding Variant of Upward Point-Set Embedding b d d a c G c c Left Page Right Page d S b b The vertices on the spine Only ordering of the vertices are important, not their positions a a The edges on the pages Spine

12. Our Algorithm 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

13. Our Algorithm • The drawing ….. • has no crossings since • it has the same embedding • as the original graph • has no spine crossings • has 1 bend per edge G contains directed hamiltonian path G contains directed hamiltonian path 7 7 6 6 5 5 4 4 3 3 2 1 2 1

14. Our Algorithm G contains no directed hamiltonian path 7 6 5 e 4 b d c 3 2 a 1

15. Our Algorithm G contains no directed hamiltonian path 7 6 5 e 4 b d c 3 2 a 1

16. Our Algorithm 7 7 6 6 5 e 5 e 4 4 d b d c c 3 b 2 a 3 1 a 2 1

17. 7 7 6 6 5 5 e e 4 4 b b d d c c 3 3 2 2 a a 1 1 Our Algorithm 7 6 5 e 4 d c b 3 a 2 1

18. 7 6 5 e 4 b d c 3 2 a 1 Our Algorithm 7 6 5 e Each spine crossing corresponds to a dummy vertex 4 d c b 3 a 2 1

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

20. e b d c a • Giordano et. al. showed that • G admits an upward point-set embedding on • S of points with the mapping φ with t bends • if and only if there is a line L such that • G admits an upward topological book embedding • with the ordering induced by φ on L with t bends Algorithm for Points in General Position e L b c S e’ φ b’ d c’ a G d’ Ordering induced by φ on L a’

21. Open Problems Find the minimum number of total bends in all edges To give an o(n3)-time testing algorithm

22. Thank You