Topological Drawings of Complete Bipartite Graphs Jean Cardinal - - PowerPoint PPT Presentation

Topological Drawings of Complete Bipartite Graphs

Jean Cardinal (ULB, Brussels) Joint work with Stefan Felsner (TU Berlin)

June 18

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Topological Drawings of Graphs

vertices ↔ points edges ↔ (well-behaved) continuous curves

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Simple Topological Drawings of Graphs

vertices ↔ points edges ↔ (well-behaved) continuous curves crossing pairwise at most once

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Simple Topological Drawings of Complete Graphs

2 1 3 4 π(v) = 2134 v Rotation system ↔ crossing edges (Pach-Tóth 06)

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Abstract Topological Graphs

G = (V, E, C), with C ⊆ E

2

• pairs of crossing edges

Simple realizability of complete AT-graphs decidable in polynomial time (Kyncl 11/15)

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Topological Drawings of Complete Bipartite Graphs

Turán’s brick factory problem Zarankiewicz’s conjecture

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Outer Drawings of Kk,n

1 previous requirement of simple topological drawings

and

2 the k vertices of one side of the bipartition lie on the

• uter boundary of the drawing.

Combinatorics of such drawings? Relevant combinatorial description and realizability checking?

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Examples

1 3 2 5 4 2 1 4 3 5 1 3 2 5 4 2 1 4 3 5 1 2 3 4 5 1 2 3 4 5 Outer drawings of K3,5 with rotation system (12345, 21435, 13254)

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A first simple case

k = 2 and uniform rotation system 1 2 3 4 4 3 2 1

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Encoding of K2,2 subdrawings

a b b a A B

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Example

1 2 3 4 4 3 2 1 1 B B B 2 B A 3 A 4

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Consistency constraints

a A B b A c

is not realizable

a b c a b c

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Triples are not enough

Only legal triples, but not realizable:

a B A B b A A c B d

a b c d a b c d

Drawings of K2,4 yield legal quadruples

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Triple and quadruple rules

a X Y b X c

⇒ Y = X

a X Y b X X c d

⇒ Y = X

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Consistency for k = 2 and uniform rotation system

Theorem Triple and quadruple consistency is sufficient for the existence of outer drawings of K2,n with uniform rotation system.

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Structure

Bijection with separable permutations = {2413, 3142}-avoiding permutations : triple rule ⇔ permutation quadruple rule ⇔ pattern avoidance Proof: consider the A, B matrices as matrices of inversions

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Arbitrary k and arbitrary rotation system

Generalization of the triple and quadruple rules Consider subdrawings of K3,2 as well Sufficiency

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Encoding of K2,2 subdrawings

a b b a a b N A B

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Triple rule

17 drawings of K2,3 – legal triples 15 triples of the form

a X Y b Z c

with Y ∈ {X, Z} 2 additional triples

a N A b B c

and

a A B b N c

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Quadruple rule

a A|B A X b A A c d

⇒ X = A

a A|B B X b B B c d

⇒ X = B

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Drawings of K3,2

1 2 1 2 2 1 1 2 1 2 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 B1 W1 B2 B3 W2 W3

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Drawings of K3,2: projections

B1 B2 B3 W1 W2 W3 T1 B A A A N N T2 A B A N A N T3 A A B N N A

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Consistency for arbitrary k

Theorem Consistency on subdrawings of K2,3 (triples), K2,4 (quadruples), and K3,2 is sufficient for the existence of

• uter drawings of Kk,n.

Corollary Outer realizability of complete bipartite AT-graphs is in P

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Proof steps

k = 2 and arbitrary rotation system k = 3 and arbitrary rotation system : case analysis Generalize from k = 3 to arbitrary k

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Other results

Rotation systems of extendable (aka pseudolinear)

• uter drawings

↔ suballowable sequences (Asinowski 2008)

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Thank you!

arXiv:1608.08324 To appear in Journal of Computational Geometry (JoCG)

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