valentin polishchuk helsinki institute for information technology, university of helsinki joint work...
TRANSCRIPT
![Page 1: Valentin Polishchuk Helsinki Institute for Information Technology, University of Helsinki Joint work with Esther Arkin and Joseph Mitchell Applied Math](https://reader035.vdocument.in/reader035/viewer/2022062515/56649d015503460f949d3e6b/html5/thumbnails/1.jpg)
Valentin Polishchuk Helsinki Institute for Information Technology, University of Helsinki
Joint work with Esther Arkin and Joseph Mitchell
Applied Math and Statistics, Stony Brook University
Two New Classes of Hamiltonian Graphs
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Induced Graph
Subset S of R2
– vertices: S
– edge (i,j) if |i – j | = 1
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Square Grid Graph
• Subset S of Z2
• Solid grid– no “holes”– all bounded faces –
unit squares
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Hamiltonicity of Square Grids
• NP-complete in general [Itai, Papadimitriou, and Szwarcfiter ’82]
• Solid grids– polynomial
[Umans and Lenhart ’96]
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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
|i – j | = 1
Hole:bounded face ≠ unit equilateral ∆
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Solid Triangular Grid
No holesall bounded faces –
unit equilateral triangles
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Previous Work
• HamCycle Problem– NP-complete in general
• Solid grids– always Hamiltonian
• no deg-1 vertices
The only non-Hamiltonian solid triangular grid
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Local CutSingle vertexwhose removaldecreases number of holes
Solid ) No local cuts
Our result:Triangular grids without
local cuts are Hamiltonian
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Idea
• B:– Cycle around the outer boundary– Cycles around holes’ boundaries
• Use modifications– cycles go through all internal vertices
• Exists “facing” rhombus– no local cuts = graph is “thick”– merge facing cycles
• Decrease number of cycles• Get Hamiltonian Cycle
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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 BL cannot be applied
s
How is s visited?
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Sharp Wedge
Z
Vs
s
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Wide Wedge
L cannot be applied t is in B
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Deja Vu
s
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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Cycle Cover → HamCycle
• Cycles around the outer boundary• Cycles around holes’ boundaries• Use modifications
– cycles go through all internal vertices
• Exists “facing” rhombus– no local cuts = graph is “thick”– merge facing cycles
• Decrease number of cycles• Get Hamiltonian Cycle
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Hamiltonian Cycles in
High-Girth Graphs
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HamCycle Problem is NP-complete
• Classic • Girth?
– 4 [GJ]
– 3 [CLRS]
• NP-complete [Garey, Johnson, Tarjan’76]
– planar– cubic– girth-5
Higher girth?
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Multi-Hamiltonicity• 1 HC 2 HCs
cubic [Smith], any vert – odd-deg [Thomason’78]
r-regular, r > 300 [Thomassen’98], r > 48 [Ghandehari and Hatami]
4-regular? conjecture [Sheehan’75]
maxdeg ≥ f( maxdeg/mindeg ) [Horak and Stacho’00]
bipartite, mindeg in a part = 3 [Thomassen’96]
• 1 HC exp(maxdeg) HCs [Thomassen’96]
– bipartite
• 1 HC exp(girth) HCs [Thomassen’96]
cubic or bipartite, mindeg in a part = 4
Planar maxdeg 3, high-girth?
>1 HC? Small # of HCs?
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Our Contribution
Planar
maxdeg 3
arbitrarily large girth
• HamCycle Problem is NP-complete
• Exactly 3 HamCycles arbitrarly large # of vertices
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The Other Tiling: Infinite Hexagonal Grid
• Induced graphs– hexagonal grids
Is HamCycle Problem NP-hard for hexagonal grids?
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Attempt to Show NP-Hardness• Same idea as for square and triangular grids
[Itai, Papadimitriou, and Szwarcfiter ‘82, Papadimitriou and Vazirani ’84, PAM’06]
• HamCycle Problem– undirected planar bipartite graphs
– max deg 3
G0
Embed
0o, 60o, 120o segments
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(Try to) Embed in Hex Grid
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Edges – Tentacles
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Traversing Tentacles
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Cross pathconnects adjacent nodes
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Return path returns to one of the nodes
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White Node Gadget
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Middle Vertex: 2 edges…
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Middle Vertex: 2 edges…
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Induces 2 cross, 1 return path
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Induces 2 cross, 1 return path
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Induces 2 cross, 1 return path
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Black Node Gadget
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Middle Vertex: 2 edges…
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Middle Vertex: 2 edges…
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Induces 2 cross, 1 return path
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Induces 2 cross, 1 return path
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Induces 2 cross, 1 return path
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Return Path Starts at white node Closes at black node
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HC in G HC in G0
Any node gadgetadjacent to
2 cross paths
1 return path
• Edges of G0 in HC
Cross paths
• Edges of G0 not in HC
Return paths from white nodes
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No… didn’t show how to turn a tentacle
Can’t turn with these tentacles
Ham Cycle is NP-hard for Hex Grid?
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No Longer in a Hex Grid
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Subdivide (Shown) Edges
Imagine:adjacent deg-2 vertices
connected by length-g path
Girth g
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Girth g+2 Graph
• Planar– turning tentacle
• no longer an issue– not in a hex grid
• Maxdeg 3
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HC in G HC in G0
Any node gadgetadjacent to
2 cross paths
1 return path
• Edges of G0 in HC
Cross paths
• Edges of G0 not in HC
Return paths from white nodes
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Theorem 1
For any g ≥ 6
HamCycle is NP-hard inplanar
deg ≤ 3non-bipartite
girth-ggraphs
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Multi-Hamiltonicity
• Planar
• Bipartite
• Maxdeg 3
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Exactly 3 HamCycles
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Theorem 2
For any g ≥ 6
existsplanar
deg ≤ 3non-bipartite
girth-ggraph
with exactly 3 HamCycles
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Summary
• Trangular grids no local cut ) Hamiltonian
• maxdeg-3 planar girth-g– HamCycle Problem is NP-complete– exists graphs with exactly 3 HamCycles
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Open
• HamCycle Problem in hexagonal grids
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