Removing Independently Even Crossings Michael Pelsmajer IIT Chicago Marcus Schaefer DePaul University Daniel Štefankovič University of Rochester
Jan 28, 2016
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)) ?