Odd Crossing Number is NOT Crossing Number
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Odd Crossing Numberis NOT
Crossing Number
Michael PelsmajerIIT (Chicago)
Marcus SchaeferDePaul University (Chicago)
Daniel ŠtefankovičUniversity of Rochester
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Crossing number
cr(G) = minimum number of crossings in a planar drawing of G
cr(K5)=?
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Crossing number
cr(G) = minimum number of crossings in a planar drawing of G
cr(K5)=1
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Rectilinear crossing number
rcr(G) = minimum number of crossings in a planar straight-line drawing of G
rcr(K5)=?
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Rectilinear crossing number
rcr(G) = minimum number of crossings in a planar straight-line drawing of G
rcr(K5)=1
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Rectilinear crossing number
rcr(G) = minimum number of crossings in a planar straight-line drawing of G
cr(G) rcr(G)
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cr(G)=0 rcr(G)=0
Every planar graph has a straight-line planar drawing.
Steinitz, Rademacher 1934 Wagner 1936Fary 1948Stein 1951
THEOREM [SR34,W36,F48,S51]:
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cr(G)=0 , rcr(G)=0
cr(G)=rcr(G)?
cr(G)=1 , rcr(G)=1cr(G)=2 , rcr(G)=2cr(G)=3 , rcr(G)=3
Are they equal?
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cr(G) rcr(G)
cr(K8) =18rcr(K8)=19
THEOREM [Guy’ 69]:
cr(G)=rcr(G)
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cr(G) rcr(G)
cr(K8) =18rcr(K8)=19
THEOREM [Guy’ 69]:
THEOREM [Bienstock,Dean ‘93]:
(8k)(9G) cr(G) =4 rcr(G)=k
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cr(G) = minimum number of crossings in a planar drawing of G
rcr(G) = minimum number of crossings in a planar straight-line drawing of G
Crossing numbers
(G) cr(G) rcr(G)
cr(G) rcr(G)
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Odd crossing number
ocr(G) = minimum number of pairs of edges crossing odd number of times
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Odd crossing number
ocr(G) = minimum number of pairs of edges crossing odd number of times
ocr(G) cr(G)
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Odd crossing number
ocr(G) = minimum number of pairs of edges crossing odd number of times
ocr(K5)=?
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Proof (Tutte’70): ocr(K5)=1
How many pairs of non-adjacentedges intersect (mod 2) ?
INVARIANT:
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Proof (Tutte’70): ocr(K5)=1
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Proof: ocr(K5)=1
How many pairs of non-adjacentidges intersect (mod 2) ?
steps which change isotopy:
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Proof: ocr(K5)=1
How many pairs of non-adjacentidges intersect (mod 2) ?
steps which change isotopy:
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Proof: ocr(K5)=1
How many pairs of non-adjacentidges intersect (mod 2) ?
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Proof: ocr(K5)=1
How many pairs of non-adjacentidges intersect (mod 2) ?
QED
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Hanani’34,Tutte’70:
ocr(G)=0 cr(G)=0
If G has drawing in which all pairs ofedges cross even # times graph is planar!
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ocr(G)=0 , cr(G)=0
ocr(G)=cr(G)?
Are they equal?
QUESTION [Pach-Tóth’00]:
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ocr(G)=0 cr(G)=0
ocr(G)=cr(G)?
Are they equal?
Pach-Tóth’00:
cr(G) 2ocr(G)2
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THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G) cr(G)
Main result
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THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G) cr(G)
1. Find G.2. Draw G to witness small ocr(G).3. Prove cr(G)>ocr(G).
How to prove it?
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THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G) cr(G)
Obstacle: cr(G)>x is co-NP-hard!
1. Find G.2. Draw G to witness small ocr(G).3. Prove cr(G)>ocr(G).
How to prove it?
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Crossing numbers for “maps”
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Crossing numbers for “maps”
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Crossing numbers for “maps”
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Ways to connect
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Ways to connect
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Ways to connect
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Ways to connect
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Ways to connect
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-1 0 +1
Ways to connect
number of “Dehn twists”
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Ways to connect
How to compute # intersections ?
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0 1 2
Ways to connect
How to compute # intersections ?
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Crossing number
min i<j|xi-xj+(i>j)|
xi2Z
the number of twists of arc i
do arcs i,j intersect in the initial drawing?
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Crossing number
min i<j|xi-xj+(i>j)|
xi2Z
the number of twists of arc i
do arcs i,j intersect in the initial drawing?
i
j
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Crossing number
min i<j|xi-xj+(i>j)|
xi2Z
the number of twists of arc i
do arcs i,j intersect in the initial drawing?
ij
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Crossing number
min i<j|xi-xj+(i>j)|
xi2Z
xi2R
OPT
OPT*
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Crossing number
min i<j|xi-xj+(i>j)|
xi2Z
xi2R
Lemma: OPT* = OPT.
OPT
OPT*
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Crossing number
min i<j|xi-xj+(i>j)|
Lemma: OPT* = OPT.
Obstacle: cr(G)>x is co-NP-hard!
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Crossing number
min i<j|xi-xj+(i>j)|
Obstacle: cr(G)>x is co-NP-hard!
yij¸ xi-xj+(i>j)
yij¸ –xi+xj-(i>j)
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Crossing number
min i<j yij
Obstacle: cr(G)>x is co-NP-hard!
yij¸ xi-xj+(i>j)
yij¸ –xi+xj-(i>j)
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Crossing number
max i<j Qij(i>j)
QT=-QQ1=0-1 Qij 1
Dual linear program
Q is an n£n matrix
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EXAMPLE:a
c b
d
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Odd crossing number ?a
c b
d
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Odd crossing numbera
c b
d
ocr ad+bc
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Crossing number ?
max i<j Qij(i>j)
a
c b
da b c d a+c d
(2,1,4,3)
cr ac+bd 0 ac b(d-a) * -ac 0 ab a(c-b)b(a-d) -ab 0 bd * a(b-c) -bd 0
QT=-QQ1=0-1 Qij 1
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Putting it together a
c b
d
ocr ad+bc
cr ac+bd
a b c d a+c d
b=c=1, a=(√3-1)/2~0.37, d=a+c
ocr/cr=√3/2~0.87
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Crossing number a
c b
d
ocr/cr=√3/2~0.87
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Crossing number a
c b
d
ocr/cr=√3/2~0.87
for graphs?
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Crossing number a
c b
d
ocr/cr=√3/2~0.87
cr=?
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Crossing number a
c b
d
ocr/cr=√3/2~0.86
cr=?
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Crossing number for graphs
ocr/cr √3/2+.
Theorem: (8 >0) (9 graph) with
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Is cr=O(ocr)?
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Is cr=O(ocr)?
Is cr O(ocr) on annulus?
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Is cr=O(ocr)?
Is cr O(ocr) on annulus?
Theorem:
On annulus cr 3ocr
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BAD triple GOOD triple
Theorem:
On annulus cr 3ocr
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BAD triple
n.CR 3#BADp
Pay: #of badtriples {p,i,j}
Average over p.
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BAD triple
#BAD n.OCR
random i,j,kX=#odd pairs
E[X] #BADbin(n,3)
3OCR bin(n,2)
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BAD triplen.CR 3#BAD#BAD n.OCR
CR 3OCR
(on annulus)
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Crossing number for graphs
ocr/cr √3/2+.There exists graph with
On annulus ocr/cr 1/3
Experimental evidence: ocr/cr √3/2 on annulus and pair of pants
Bold (wrong) conjecture: For any graph ocr/cr √3/2
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Questions
crossing number of mapswith d vertices in poly-time?(true for d 2)
(map = graph + rotation system)
Bold (wrong) conjecture: For any graph ocr/cr √3/2
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Open questions - classic
Zarankiewicz’s conjecture:
cr(Km,n)
Guy’s conjecture:
cr(Kn)
Better approx algorithm for cr.
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Crossing number for graphs
pair crossing number (pcr) # number of pairs of crossing edges
algebraic crossing number (acr) algebraic crossing number of edges
+1
-1
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Crossing numbers
ocr(G) acr(G)
pcr(G)
cr(G) rcr(G)