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Odd Crossing Number is NOT Crossing Number. Michael Pelsmajer IIT (Chicago) Marcus Schaefer DePaul University (Chicago) Daniel Štefankovič University of Rochester. Crossing number. cr(G) = minimum number of crossings in a planar drawing of G. cr(K 5 )=?. Crossing number.
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Odd Crossing Number is NOT Crossing Number Michael Pelsmajer IIT (Chicago) Marcus Schaefer DePaul University (Chicago) Daniel Štefankovič University of Rochester
Crossing number cr(G) = minimum number of crossings in a planar drawing of G cr(K5)=?
Crossing number cr(G) = minimum number of crossings in a planar drawing of G cr(K5)=1
Rectilinear crossing number rcr(G) = minimum number of crossings in a planarstraight-line drawing of G rcr(K5)=?
Rectilinear crossing number rcr(G) = minimum number of crossings in a planarstraight-line drawing of G rcr(K5)=1
Rectilinear crossing number rcr(G) = minimum number of crossings in a planarstraight-line drawing of G cr(G) rcr(G)
cr(G)=0 rcr(G)=0 THEOREM [SR34,W36,F48,S51]: Every planar graph has a straight-line planar drawing. Steinitz, Rademacher 1934 Wagner 1936 Fary 1948 Stein 1951
Are they equal? cr(G)=0 , rcr(G)=0 cr(G)=1 , rcr(G)=1 cr(G)=2 , rcr(G)=2 cr(G)=3 , rcr(G)=3 ? cr(G)=rcr(G)
cr(G) rcr(G) THEOREM [Guy’ 69]: cr(K8) =18 rcr(K8)=19 cr(G)=rcr(G)
cr(G) rcr(G) THEOREM [Guy’ 69]: cr(K8) =18 rcr(K8)=19 THEOREM [Bienstock,Dean ‘93]: (8k)(9G) cr(G) =4 rcr(G)=k
Crossing numbers cr(G) = minimum number of crossings in a planar drawing of G rcr(G) = minimum number of crossings in a planarstraight-line drawing of G cr(G) rcr(G) (G) cr(G) rcr(G)
Odd crossing number ocr(G) = minimum number of pairs of edges crossing odd number of times
Odd crossing number ocr(G) = minimum number of pairs of edges crossing odd number of times ocr(G) cr(G)
Odd crossing number ocr(G) = minimum number of pairs of edges crossing odd number of times ocr(K5)=?
Proof (Tutte’70): ocr(K5)=1 INVARIANT: How many pairs of non-adjacent edges intersect (mod 2) ?
Proof: ocr(K5)=1 How many pairs of non-adjacent idges intersect (mod 2) ? steps which change isotopy:
Proof: ocr(K5)=1 How many pairs of non-adjacent idges intersect (mod 2) ? steps which change isotopy:
Proof: ocr(K5)=1 How many pairs of non-adjacent idges intersect (mod 2) ?
Proof: ocr(K5)=1 How many pairs of non-adjacent idges intersect (mod 2) ? QED
Hanani’34,Tutte’70: ocr(G)=0 cr(G)=0 If G has drawing in which all pairs of edges cross even # times graph is planar!
Are they equal? ocr(G)=0 , cr(G)=0 QUESTION [Pach-Tóth’00]: ? ocr(G)=cr(G)
Are they equal? ocr(G)=0 cr(G)=0 ? ocr(G)=cr(G) Pach-Tóth’00: cr(G) 2ocr(G)2
Main result THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G) cr(G)
How to prove it? THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G) cr(G) • Find G. • Draw G to witness small ocr(G). • Prove cr(G)>ocr(G).
How to prove it? THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G) cr(G) • Find G. • Draw G to witness small ocr(G). • Prove cr(G)>ocr(G). Obstacle: cr(G)>x is co-NP-hard!
Ways to connect number of “Dehn twists” -1 0 +1
Ways to connect How to compute # intersections ?
Ways to connect How to compute # intersections ? 0 2 1
Crossing number min i<j|xi-xj+(i>j)| xi2Z do arcs i,j intersect in the initial drawing? the number of twists of arc i
Crossing number i min i<j|xi-xj+(i>j)| xi2Z j do arcs i,j intersect in the initial drawing? the number of twists of arc i
Crossing number min i<j|xi-xj+(i>j)| xi2Z j i do arcs i,j intersect in the initial drawing? the number of twists of arc i
Crossing number min i<j|xi-xj+(i>j)| OPT xi2Z OPT* xi2R
Crossing number min i<j|xi-xj+(i>j)| OPT xi2Z OPT* xi2R Lemma: OPT* = OPT.
Crossing number min i<j|xi-xj+(i>j)| Lemma: OPT* = OPT. Obstacle: cr(G)>x is co-NP-hard!
Crossing number min i<j|xi-xj+(i>j)| yij¸ xi-xj+(i>j) yij¸ –xi+xj-(i>j) Obstacle: cr(G)>x is co-NP-hard!
Crossing number min i<j yij yij¸ xi-xj+(i>j) yij¸ –xi+xj-(i>j) Obstacle: cr(G)>x is co-NP-hard!
Crossing number Dual linear program max i<j Qij(i>j) QT=-Q Q1=0 -1 Qij 1 Q is an n£n matrix
EXAMPLE: a b c d
Odd crossing number ? a b c d
Odd crossing number a ocr ad+bc b c d
Crossing number ? a max i<j Qij(i>j) b QT=-Q Q1=0 -1 Qij 1 c d =(2,1,4,3) a b c d a+c d 0 ac b(d-a) * -ac 0 ab a(c-b) b(a-d) -ab 0 bd * a(b-c) -bd 0 cr ac+bd