Hamiltonian Cycles in Triangular Grids

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Hamiltonian Cycles in Triangular Grids

• Valentin Polishchuk
• joint work with
• Estie Arkin and Joseph Mitchell

Applied Math and Statistics Stony Brook University
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Grids
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Grids, grids
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Grids, grids, grids
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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

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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

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Tilings

• Square grid
• unit squares

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Tilings

• Square grid
• unit squares

• Triangular grid
• unit equilateral triangles

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Triangular Grid Graph

• Subset S
• vertices S
• edge (i,j) if
• Si Sj 1

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Solid Triangular Grid

• No holes
• all bounded faces
• unit equilateral triangles

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Grids
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Grids, grids
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Hamiltonicity of Grids
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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?

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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

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NP-Completeness Results
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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
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The Node Gadget
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The Tentacles
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Connection to a White Node
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Connection to a Black Node
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Traversing Tentacles

• Black node-tentacle connection
• a cut

Return path Cross path returns to
white node connects white and black nodes
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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

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Max degree 4 Grids

• No degree-6 vertices
• Degree-5 vertices

Modified gadgets
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Almost AllSolid Grids are Hamiltonian
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Assumption

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

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Boundary

• deg lt 6 vertices
• Internal vertices
• deg-6 vertices
• No cut vertex
• cycle B around the boundary
• visits every bd vertex once

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Idea

• B cycle around the boundary
• Local modifications
• attach to B internal vertices
• cost 1 per vertex

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L-modification
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V-modification
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Z-modification
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Priority L , V , Z

• L
• V
• Z

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Wedges

• Sharp
• 60o turn
• Wide
• 120o turn

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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

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Internal vertex v not in B

• A neighbor u is in B
• Crossed edges
• not in B
• o.w. apply L

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How is u visited?

• WLOG, 1 is in B

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s is in B
L cannot be applied
How is s visited?
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Sharp Wedge
s
V

• Z

s
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Wide Wedge

• L cannot be applied t is in B

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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

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Another Wide Wedge

• Yet Another vertex
• Yet Another rhombus

Yet Another wide wedge
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And so on

• Star of David!

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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

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Extensions

• Grids with holes
• no local cut
• Manifolds (meshes)
• some assumptions

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Open

• Max degree 3
• Hexagonal grids

Thank you!