removing independently even crossings
DESCRIPTION
Removing Independently Even Crossings. Michael Pelsmajer IIT Chicago Marcus Schaefer DePaul University Daniel Štefankovič University of Rochester. Crossing number. cr(G) = minimum number of crossings in a drawing * of G. cr(K 5 )=1. - PowerPoint PPT PresentationTRANSCRIPT
Removing IndependentlyEven Crossings
Michael PelsmajerIIT Chicago
Marcus SchaeferDePaul University
Daniel ŠtefankovičUniversity of Rochester
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 don’t cross vertices, edges do not touch)
Crossing number
● don’t know cr(Kn), cr(Km,n)
Zarankiewicz’s conjecture:
cr(Km,n)=
Guy’s conjecture:
cr(Kn)=
● no approximation algorithm
poorly understood, for example:
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 don’t cross vertices, edges do not touch)
Odd crossing number
ocr(G) = minimum number of pairs of edges that cross oddly in a drawing* of G
ocr(K5)=1
*(general position drawings, i.e., no intersections with 3 edges, edges don’t cross vertices, edges do not touch)
oddly = odd number of times
Rectilinear crossing number
rcr(G) = minimum number of crossings in a planar straight-line drawing of G
rcr(K5)=1
“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
“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
iocr(G)=CVP {e0,e1}
(v,g)
1 if g=ei and v is an endpoint of e1-i
0 otherwise
any initial drawing
columns = pair of non-adjacent edges, e.g., for K5, 15 columnsrows = non-adjacent (vertex,edge), e.g., for K5, 30 rows
iocr(G)=CVP {e0,e1}
(v,g)
1 if g=ei and v is an endpoint of e1-i
0 otherwise
any initial drawing
columns = pair of non-adjacent edges, e.g., for K5, 15 columnsrows = non-adjacent (vertex,edge), e.g., for K5, 30 rows
[ ], , , , , , , , , , , , , ,0 1 0 0 1 0 0 1 1 0 1 0 0 0 0
Crossing numbers
ocr(G)
acr(G)
pcr(G)
cr(G) rcr(G)
iocr(G)
0100
0102
0111
0122
ocr acr pcr cr
Crossing numbers – amazing fact
iocr(G)=0 rcr(G)=0
ocr(G)
acr(G)
pcr(G)
cr(G) rcr(G)
iocr(G)
iocr(G)=0 cr(G)=0 (Hanani’34,Tutte’70)
cr(G)=0 rcr(G)=0 (Steinitz, Rademacher’34; Wagner ’36; Fary’48; Stein’51)
Crossing numbers – amazing fact
iocr(G) 2 rcr(G)=iocr(G)
ocr(G)
acr(G)
pcr(G)
cr(G) rcr(G)
iocr(G)
iocr(G) 2 cr(G)=iocr(G) (present paper)
cr(G) 3 rcr(G)=cr(G) (Bienstock, Dean’93)
Crossing numbers - separation
ocr(G)
acr(G)
pcr(G)
cr(G) rcr(G)
Pelsmajer, Schaefer, Štefankovič’05
Tóth’08
Guy’69
different
maybe equal?
iocr(G)
cr(K8) =18,rcr(K8)=19
Crossing numbers - separation
ocr(G)
acr(G)
pcr(G)
cr(G) rcr(G)
different
maybe equal?
iocr(G)BIG
Bienstock,Dean ’93( k 4)(G)cr(G)=4, rcr(G)=k
very different
Crossing numbers - separation
ocr(G)
acr(G)
pcr(G)
cr(G) rcr(G)
different
maybe equal?
iocr(G)BIG
very different
polynomially relatedPach, Tóth’00
cr(G) ( )2ocr(G)2
Bienstock,Dean ’93( k 4)(G)cr(G)=4, rcr(G)=k
Crossing numbers - separation
ocr(G)
acr(G)
pcr(G)
cr(G) rcr(G)
different
maybe equal?
iocr(G)
very different
polynomially relatedPach, Tóth’00
cr(G) ( )2ocr(G)2
Bienstock,Dean ’93( k 4)(G)cr(G)=4, rcr(G)=k
our result
cr(G) ( )2iocr(G)2
BIG
different
very differentour result
cr(G) ( )2iocr(G)2
drawing D realizing iocr(G)
bad edges
good edges
|bad|2iocr(G)
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)
GOAL: drawing D’ such that • good edges are intersection free• pair of bad edges intersects 1 times
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
even edges
even edges
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
cycle C consisting of even edges
redrawing so that C is intersection free, no new odd pairs, same rotation system
Lemma (Pelsmajer, Schaefer, Stefankovic’07)
good even, locally
bad edges
good edges
|bad|2iocr(G)
even edges
cycle of good edges cycle of even edges intersection free cycle
good even, locally
bad edges
good edges
|bad|2iocr(G)
even edges
cycle of good edges cycle of even edges intersection free cycle
good even, locally
bad edges
good edges
|bad|2iocr(G)
even edges
cycle of good edges cycle of even edges intersection free cycle
good even, locally
bad edges
good edges
|bad|2iocr(G)
even edges
cycle of good edges cycle of even edges intersection free cycle degree 3 vertices
good even, locally
bad edges
good edges
|bad|2iocr(G)
even edges
cycle of good edges cycle of even edges intersection free cycle degree 3 vertices
good even, locallycycle of good edges cycle of even edges intersection free cycle degree 3 vertices
repeat, repeat, repeat
= dv
3#good cycles with intersections
potentials decreasing:
DONE good edges in cycles are intersection free
bad edges
good edges
DONE good edges in cycles are intersection free
good edgesnot in a good cycle
bad edges
good edges
look at the blue faces
good edgesnot in a good cycle
bad edges
good edges
add violet good edges, no new faces
good edgesnot in a good cycle
bad edges
good edges
add bad edges in their faces ...
good edgesnot in a good cycle
Open problems
Is pcr(G)=cr(G) ?
A
A
B
B C
C
D
D
on annulus?
Open problems
Is iocr(G)=ocr(G) ?
Does iocrg(G)=0 crg(G)=0 ?(genus g strong Hannani-Tutte)
Is cr(G)=O(iocr(G)) ?