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)) ?