Removing Independently

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Removing Independently Even Crossings
Michael Pelsmajer IIT Chicago Marcus
Schaefer DePaul University Daniel
Štefankovic University of Rochester
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Crossing number
cr(G) minimum number of crossings
in a drawing of G
cr(K5)1
(general position drawings, i.e., no
intersections with 3 edges, edges dont cross
vertices, edges do not touch)
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Crossing number
poorly understood, for example
? dont know cr(Kn), cr(Km,n)
Guys conjecture
cr(Kn)
Zarankiewiczs conjecture
cr(Km,n)
? no approximation algorithm
4
Pair crossing number
pcr(G) minimum number of pairs of
edges that cross in a drawing of G
pcr(K5)1
(general position drawings, i.e., no
intersections with 3 edges, edges dont cross
vertices, edges do not touch)
5
Odd crossing number
ocr(G) minimum number of pairs of
edges that cross oddly in a
drawing of G
ocr(K5)1
oddly odd number of times
(general position drawings, i.e., no
intersections with 3 edges, edges dont cross
vertices, edges do not touch)
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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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Independent crossing numbers
only non-adjacent edges contribute
iocr(G)minimum number of pairs of
non-adjacent edges that cross oddly
in a drawing of G
ocr(G) minimum number of pairs of
edges that cross oddly in a drawing
of G
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Independent crossing numbers
only non-adjacent edges contribute
iocr(G)minimum number of pairs of
non-adjacent edges that cross oddly
in a drawing of G
What should be the ordering of edges around
v? independent ? does not matter!
v
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e0,e1
iocr(G)CVP
any initial drawing
(v,g)
1 if gei and v is an endpoint of e1-i 0
otherwise
columns pair of non-adjacent edges, e.g., for
K5, 15 columns rows non-adjacent
(vertex,edge), e.g., for K5, 30 rows
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e0,e1
iocr(G)CVP
any initial drawing
(v,g)
1 if gei and v is an endpoint of e1-i 0
otherwise
columns pair of non-adjacent edges, e.g., for
K5, 15 columns rows non-adjacent
(vertex,edge), e.g., for K5, 30 rows
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Crossing numbers
iocr(G)
acr(G)
?
?
?
cr(G)
rcr(G)
ocr(G)
?
?
?
pcr(G)
ocr
acr
cr
pcr
0 1 0 0
0 1 0 2
0 1 1 1
0 1 2 2
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Crossing numbers amazing fact
iocr(G)
acr(G)
?
?
?
ocr(G)
cr(G)
rcr(G)
?
?
?
pcr(G)
iocr(G)0 ? rcr(G)0
iocr(G)0 ? cr(G)0 (Hanani34,Tutte70) cr(G)0
? rcr(G)0 (Steinitz, Rademacher34 Wagner
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Fary48 Stein51)
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Crossing numbers amazing fact
iocr(G)
acr(G)
?
?
?
ocr(G)
cr(G)
rcr(G)
?
?
?
pcr(G)
iocr(G) ? 2 ? rcr(G)iocr(G)
iocr(G) ? 2 ? cr(G)iocr(G) (present
paper) cr(G) ? 3 ? rcr(G)cr(G) (Bienstock,
Dean93)
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Crossing numbers - separation
iocr(G)
acr(G)
ocr(G)
cr(G)
rcr(G)
pcr(G)
Guy69
cr(K8) 18, rcr(K8)19
Tóth08
Pelsmajer, Schaefer, Štefankovic05
different
maybe equal?
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Crossing numbers - separation
BIG
iocr(G)
acr(G)
ocr(G)
cr(G)
rcr(G)
pcr(G)
Bienstock,Dean 93 (? k ? 4)(?G) cr(G)4, rcr(G)k
very different
different
maybe equal?
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Crossing numbers - separation
BIG
iocr(G)
acr(G)
ocr(G)
cr(G)
rcr(G)
pcr(G)
Bienstock,Dean 93 (? k ? 4)(?G) cr(G)4, rcr(G)k
polynomially related
Pach, Tóth00 cr(G) ?
( )
very different
2ocr(G)
2
different
maybe equal?
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Crossing numbers - separation
BIG
iocr(G)
acr(G)
ocr(G)
cr(G)
rcr(G)
pcr(G)
Bienstock,Dean 93 (? k ? 4)(?G) cr(G)4, rcr(G)k
polynomially related
Pach, Tóth00 cr(G) ?
very different
( )
our result cr(G) ?
2ocr(G)
2
( )
2iocr(G)
different
2
maybe equal?
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e is bad if ?f such that ? e,f independent ?
e,f cross oddly
drawing D realizing iocr(G)
bad edges
good edges
bad?2iocr(G)
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drawing D realizing iocr(G)
bad edges
good edges
bad?2iocr(G)

• GOAL drawing D such that
• good edges are intersection free
• pair of bad edges intersects ? 1 times

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drawing D realizing iocr(G)
bad edges
good edges
even edges
bad?2iocr(G)

• GOAL drawing D such that
• good edges are intersection free
• pair of bad edges intersects ? 1 times

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drawing D realizing iocr(G)
bad edges
good edges
Lemma (Pelsmajer, Schaefer, Stefankovic07)
even edges
cycle C consisting of even edges
bad?2iocr(G)
redrawing so that C is intersection free, no new
odd pairs, same rotation system

• GOAL drawing D such that
• good edges are intersection free
• pair of bad edges intersects ? 1 times

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good ? even, locally
bad edges
good edges
even edges
bad?2iocr(G)
cycle of good edges ? cycle of even edges ?
intersection free cycle
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good ? even, locally
bad edges
good edges
even edges
bad?2iocr(G)
cycle of good edges ? cycle of even edges ?
intersection free cycle
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good ? even, locally
bad edges
good edges
even edges
bad?2iocr(G)
cycle of good edges ? cycle of even edges ?
intersection free cycle
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good ? even, locally
bad edges
good edges
even edges
bad?2iocr(G)
cycle of good edges ? cycle of even edges ?
intersection free cycle ? degree ?3
vertices
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good ? even, locally
bad edges
good edges
even edges
bad?2iocr(G)
cycle of good edges ? cycle of even edges ?
intersection free cycle ? degree ?3
vertices
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good ? even, locally
cycle of good edges ? cycle of even edges ?
intersection free cycle ? degree ?3
vertices
repeat, repeat, repeat
potentials decreasing
? ? dv3
good cycles with intersections
DONE ? good edges in cycles are intersection free
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DONE ? good edges in cycles are intersection free
bad edges
good edges
good edges not in a good cycle
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look at the blue faces
bad edges
good edges
good edges not in a good cycle
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add violet good edges, no new faces
bad edges
good edges
good edges not in a good cycle
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add bad edges in their faces ...
bad edges
good edges
good edges not in a good cycle
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Open problems
Is pcr(G)cr(G) ?
D
A
B
C
D
A
on annulus?
B
C
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Open problems
Is iocr(G)ocr(G) ?
(genus g strong Hannani-Tutte)
Does iocrg(G)0 ? crg(G)0 ?
Is cr(G)O(iocr(G)) ?