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Unit Contact Representationsof Grid Subgraphs with Regular Polytopes
in 2D and 3D
Linda Kleist & Benjamin RahmanTechnische Universität Berlin
Linda Kleist GD 2014: UPCRs 1
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UPCR with regular polygons
unit proper contact representation
⇐⇒ (u, v) ∈ E
(d− 1)-dimens. intersection
u
v
congruent
vertices: congruent regular polygons, interiorly disjointedges: (d−1)-dimensional intersections
Linda Kleist GD 2014: UPCRs 2
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UPCR with regular polygons
unit proper contact representation
⇐⇒ (u, v) ∈ E
(d− 1)-dimens. intersection
u
v
congruent
vertices: congruent regular polygons, interiorly disjointedges: (d−1)-dimensional intersections
Linda Kleist GD 2014: UPCRs 2
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UPCR with regular polygons
unit proper contact representation
⇐⇒ (u, v) ∈ E
(d− 1)-dimens. intersection
u
v
congruent
vertices: congruent regular polygons, interiorly disjointedges: (d−1)-dimensional intersections
Linda Kleist GD 2014: UPCRs 2
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Basic properties of UPCR
low maximal degreevolume constraints
} Ô⇒Let G be a grid. Does everysubgraph G ⊆G has a UPCR(with a particular object type)?
NP-hard recognition– unit disks [Breu, Kirkpatrick, 1996 ]– unit cubes [Bremner, Evans, Frati, Heyer, Kobourov, Lenhart, Liotta, Rappaport, Whitesides, 2013 ]– squares, (triangles, hexagons, 6k-gons, ...) [K., Rahman, 2014 ]
Linda Kleist GD 2014: UPCRs 3
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Previous Work and Questions
Theorem (Alam, Chaplick, Fijav, Kaufmann, Kobourov, Pupyrev, 2013 )
Every subgraph of the square grid allows for a UPCR with cubes.
Open:Do subgraphs of the triangular grid allow for UPCR with cubes?
Linda Kleist GD 2014: UPCRs 4
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Results with strategy
Every subgraph of has a UPCR with
4k-gons
3k-gons
square grid
d-dimen. grid
squares pseudo-squares
d-cubes
cubestriangular grid
hexagonal grid triangles pseudo-triangles
Linda Kleist GD 2014: UPCRs 5
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Strategy
Strategystart with UPCR φ̂ of the grid Gremove unwanted contacts one by one
moving setdirection vector
Linda Kleist GD 2014: UPCRs 6
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USqPCR
Square grid Sn v0,1
v1,0 . . .
⋮
vn,0
v0,n
v0,0
TheoremLet G be a subgraph of Sn. Then G has a USqPCR.
Linda Kleist GD 2014: UPCRs 7
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USqPCR
TheoremLet G be a subgraph of Sn. Then G has a USqPCR.
USqPCR φ̂ of Sn with ε ∈ (0,1)
v1,0v2,0
vn,0v0,2
v0,1
v0,n
v0,0
d2
d1
ε
E = E1∪E2 (column and row edges)
Linda Kleist GD 2014: UPCRs 8
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USqPCR
TheoremLet G be a subgraph of Sn. Then G has a USqPCR.
v1,0v2,0
vn,0v0,2
v0,1
v0,n
v0,0
d2
d1
E = E1∪E2 (column and row edges)direction vectors d(e)
Linda Kleist GD 2014: UPCRs 8
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USqPCR
TheoremLet G be a subgraph of Sn. Then G has a USqPCR.
v1,0v2,0
vn,0v0,2
v0,1
v0,n
v0,0
d2
d1
E = E1∪E2 (column and row edges)moving sets M(e)
Linda Kleist GD 2014: UPCRs 8
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Construction- more formal
ε ∈ (0,1), δ < 1n min{ε,1−ε} φ ∶V →P(R2)
φ(v) = φ̂(v)+∑i
ri(v) ⋅δdi
φ̂
φ
Propertiescs(φ̂) = 1−ε
spφ̂(M(e),d(e)) ≥ ε
cs(φ) ≥ 1−ε −nδ
ri(u) = ri(v) ⇐⇒ (u,v) ∈ E ∩Ei
interiorly disjoint (–> space)correct contacts (–> contact size)correct non-contacts (–> translation)
Linda Kleist GD 2014: UPCRs 9
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Generalization to all dimensions
TheoremLet G be a subgraph of Sd
n . Then G has a UPCR with d-cubes.
Linda Kleist GD 2014: UPCRs 10
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Generalization to all dimensions
TheoremLet G be a subgraph of Sd
n . Then G has a UPCR with d-cubes.
φ̂ ∶V →P(Rd)φ̂(vx) =Q(A ⋅x)
A ∶=⎛⎜⎝
1 ε . . . ε−ε 1 ⋱ ⋮⋮ ⋱ ⋱ ε
−ε . . . −ε 1
⎞⎟⎠
Linda Kleist GD 2014: UPCRs 10
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Generalization to all dimensions
TheoremLet G be a subgraph of Sd
n . Then G has a UPCR with d-cubes.
d types of edges: E = E1∪⋅ ⋅ ⋅∪Ed
d1
d2d3
Linda Kleist GD 2014: UPCRs 10
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Generalization to all dimensions
TheoremLet G be a subgraph of Sd
n . Then G has a UPCR with d-cubes.
d types of edges: E = E1∪⋅ ⋅ ⋅∪Ed
d1
d2d3
Linda Kleist GD 2014: UPCRs 10
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Generalization to all dimensions
TheoremLet G be a subgraph of Sd
n . Then G has a UPCR with d-cubes.
d types of edges: E = E1∪⋅ ⋅ ⋅∪Ed
d1
d2d3
Linda Kleist GD 2014: UPCRs 10
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Generalization to all dimensions
TheoremLet G be a subgraph of Sd
n . Then G has a UPCR with d-cubes.
d1
d2d3
φ ∶V →P(Rd)
φ(v) = φ̂(v)+d
∑k=1
rk(v) ⋅δdk.
Linda Kleist GD 2014: UPCRs 10
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Triangular grid
TheoremLet G be a subgraph of Tn,m. Then G has a UCuPCR.
b0,0b1,0 b4,0. . .
b1,0
b2,0
t0,0
t1,0
t2,0
Linda Kleist GD 2014: UPCRs 11
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Triangular grid
TheoremLet G be a subgraph of Tn,m. Then G has a UCuPCR.
b0,0b1,0 b4,0. . .
b1,0
b2,0
t0,0
t1,0
t2,0
Linda Kleist GD 2014: UPCRs 11
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More regular polygons
Pseudo-polygons
LemmaLet G be a graph with a UPCR φ with regular k-gons and cs(φ) > 1− s. Then, Ghas a UPCR with pseudo k-gons with side length ≥ s.
CorollaryLet G be a subgraph of Sn. Then G has a UPCR with 4k-gons(pseudo-squares).
Linda Kleist GD 2014: UPCRs 12
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More regular polygons
LemmaLet G be a graph with a UPCR φ with regular k-gons and cs(φ) > 1− s. Then, Ghas a UPCR with pseudo k-gons with side length ≥ s.
CorollaryLet G be a subgraph of Sn. Then G has a UPCR with 4k-gons(pseudo-squares).
Linda Kleist GD 2014: UPCRs 12
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More regular polygons
LemmaLet G be a graph with a UPCR φ with regular k-gons and cs(φ) > 1− s. Then, Ghas a UPCR with pseudo k-gons with side length ≥ s.
CorollaryLet G be a subgraph of Sn. Then G has a UPCR with 4k-gons(pseudo-squares).
Linda Kleist GD 2014: UPCRs 12
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More regular polygons
LemmaLet G be a graph with a UPCR φ with regular k-gons and cs(φ) > 1− s. Then, Ghas a UPCR with pseudo k-gons with side length ≥ s.
CorollaryLet G be a subgraph of Sn. Then G has a UPCR with 4k-gons(pseudo-squares).
Linda Kleist GD 2014: UPCRs 12
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More regular polygons: 3k-gons
TheoremLet G be a subgraph of Hn,m. Then G has a UPCR with 3k-gons(pseudo-triangles).
Triangles+Lemma
d2
d1
Linda Kleist GD 2014: UPCRs 13
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Open problems
1 Characterization of graphs with USqPCRs?2 Or with other polygons?
3 Is it NP-hard to recognize graphs admitting UPCRs with regular(2k+1)-gons?
4 Is it NP-hard to recognize graphs admitting UPCRs with d-cubes?
5 USqPCR for trihexagonal and truncated trihexagonal grid?6 USqPCR for dual of snubsquare grid?7 UCuPCR for duals of Archimedean grids not containing K1,9?
Thanks! ,
Linda Kleist GD 2014: UPCRs 14
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Open problems
1 Characterization of graphs with USqPCRs?2 Or with other polygons?
3 Is it NP-hard to recognize graphs admitting UPCRs with regular(2k+1)-gons?
4 Is it NP-hard to recognize graphs admitting UPCRs with d-cubes?
5 USqPCR for trihexagonal and truncated trihexagonal grid?6 USqPCR for dual of snubsquare grid?7 UCuPCR for duals of Archimedean grids not containing K1,9?
Thanks! ,Linda Kleist GD 2014: UPCRs 14
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USqPCR of Archimeadian grids
Linda Kleist GD 2014: UPCRs 15