Hamiltonian Cycles in Triangular Grids

1. Hamiltonian Cycles in Triangular Grids Valentin Polishchuk joint work with Estie Arkin and Joseph Mitchell Applied Math and Statistics Stony Brook University

2. Grids

3. Grids, grids

4. Grids, grids, grids…

5. Square Grid Graph • Subset S of Z2 • vertices: S • edge (i,j) if |Si –Sj | = 1 • Solid grid • no “holes” • all bounded faces – unit squares

6. Hamiltonicity of Square Grids • NP-compete in general [Itai, Papadimitriou, and Szwarcfiter ’82] • even if max deg = 3 [Papadimitriou and Vazirani ’84, Buro ‘03] • Solid grids • polynomial [Umans and Lenhart ’96] • short covering tour [Arkin, Fekete, Mitchell ‘92] • length (6/5)N • linear time

7. Tilings • Square grid • unit squares

8. Tilings • Square grid • unit squares • Triangular grid • unit equilateral triangles

9. Triangular Grid Graph Subset S • vertices: S • edge (i,j) if |Si –Sj | = 1

10. Solid Triangular Grid No “holes” all bounded faces – unit equilateral triangles

11. Grids

12. Grids, grids

13. Hamiltonicity of Grids

14. Previous Work • Other triangulations [Dillencourt ’92, Arkin, Held, Mitchell, Skiena ’96, Cimikowski ’90, ’93, Dogrusoz and Krishnamoorthy ’95, Flatland ‘04] • Long cycles through faces • stripification [Bushan, Diaz-Gutierrez, Eppstein, Gopi’ 04,’06] • possible subdivision • relaxed notion of Hamiltonicity [Demaine, Eppstein, Erickson, Hart, O'Rourke ‘01] Hamiltonian cycle through vertices?

15. Our Contribution • NP-compete in general • even if max deg = 4 • Solid triangular grids • polynomial • cut-free – Hamiltonian • not the Star of David • linear-time to find a cycle • linear-time solution to TSP

16. NP-Completeness Results

17. The Reduction • Same idea as for square grids [Itai, Papadimitriou, and Szwarcfiter ‘82, Papadimitriou and Vazirani ‘84] • Hamiltonian Cycle • undirected planar bipartite graphs • max deg 3 G0 Embed 0o, 60o, 120o segments

18. The Node Gadget

19. The Tentacles

20. Connection to a White Node

21. Connection to a Black Node

22. Traversing Tentacles Black node-tentacle connection • a cut Return path Cross path returns to white node connects white and black nodes

23. HC in G HC in G0 Any node gadget adjacent to 2 cross paths Return path Cross path • Edges of G0 in HC Cross paths • Edges of G0 not in HC Return paths from white nodes

24. Max degree 4 Grids • No degree-6 vertices • Degree-5 vertices Modified gadgets

25. Almost AllSolid Grids areHamiltonian

26. Assumption No cut vertex • removal disconnects G WLOG • o.w. – no cycle

27. Boundary • deg < 6 vertices • Internal vertices • deg-6 vertices • No cut vertex • cycle B around the boundary • visits every bd vertex once

28. Idea • B – cycle around the boundary • Local modifications • attach to B internal vertices • cost: 1 per vertex

29. L-modification

30. V-modification

31. Z-modification

32. Priority: L , V , Z • L • V • Z

33. Wedges • Sharp • 60o turn • Wide • 120o turn

34. The Main Lemma Until B passes through ALL internal vertices • either L, V, or Z may be applied small print: unless G is the Star of David

35. Internal vertex v not in B • A neighbor u is in B • Crossed edges • not in B • o.w. – apply L

36. How is u visited? WLOG, 1 is in B

37. s L cannot be applied s is in B How is s visited?

38. Sharp Wedge Z s V s

39. Wide Wedge L cannot be applied t is in B

40. s Deja Vu Rhombus • edge of B • vertex not in B • vertex in B Unless • t is a wide wedge • modification! • welcome new vertex to B

41. Another Wide Wedge Yet Another vertex • Yet Another rhombus Yet Another wide wedge

42. And so on… Star of David!

43. Summary • Triangular grids • solid • Hamiltonian Cycle Problem • NP-compete even if max deg = 4 • Solid triangular grids • cut-free – Hamiltonian • not the Star of David • linear-time to find a cycle • linear-time solution to TSP ”Topological” proof

44. Extensions • Grids with holes • no “local cut” • Manifolds (meshes) • some assumptions

45. Open • Max degree 3 • Hexagonal grids Thank you!