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 1 / 26

Topological Drawings of Graphs vertices ↔ points edges ↔ (well-behaved) continuous curves 2 / 26

Simple Topological Drawings of Graphs vertices ↔ points edges ↔ (well-behaved) continuous curves crossing pairwise at most once 3 / 26

Simple Topological Drawings of Complete Graphs 3 1 v 4 2 π ( v ) = 2134 Rotation system ↔ crossing edges (Pach-Tóth 06) 4 / 26

Abstract Topological Graphs � E � G = ( V , E , C ) , with C ⊆ pairs of crossing edges 2 Simple realizability of complete AT-graphs decidable in polynomial time (Kyncl 11/15) 5 / 26

Topological Drawings of Complete Bipartite Graphs Turán ’s brick factory problem Zarankiewicz ’s conjecture 6 / 26

Outer Drawings of K k , n 1 previous requirement of simple topological drawings and 2 the k vertices of one side of the bipartition lie on the outer boundary of the drawing. Combinatorics of such drawings? Relevant combinatorial description and realizability checking? 7 / 26

Examples 1 2 3 4 5 1 2 3 4 5 2 1 4 3 5 2 1 4 3 5 1 3 2 5 4 1 3 2 5 4 Outer drawings of K 3 , 5 with rotation system ( 12345 , 21435 , 13254 ) 8 / 26

A first simple case k = 2 and uniform rotation system 4 1 3 2 2 3 1 4 9 / 26

Encoding of K 2 , 2 subdrawings b a A B a b 10 / 26

Example 1 B B B 4 1 3 2 2 B A 2 3 3 A 1 4 4 11 / 26

Consistency constraints a A B b A is not realizable c c a b b a c 12 / 26

Triples are not enough Only legal triples, but not realizable: d a c b b c a B A B a d b A A c B d Drawings of K 2 , 4 yield legal quadruples 13 / 26

Triple and quadruple rules a X Y ⇒ Y = X b X c a X Y b X X ⇒ Y = X c d 14 / 26

Consistency for k = 2 and uniform rotation system Theorem Triple and quadruple consistency is sufficient for the existence of outer drawings of K 2 , n with uniform rotation system. 15 / 26

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 16 / 26

Arbitrary k and arbitrary rotation system Generalization of the triple and quadruple rules Consider subdrawings of K 3 , 2 as well Sufficiency 17 / 26

Encoding of K 2 , 2 subdrawings N b A B b a a b a 18 / 26

Triple rule 17 drawings of K 2 , 3 – legal triples 15 triples of the form a X Y b Z c with Y ∈ { X , Z } 2 additional triples a N A a A B b B b N and c c 19 / 26

Quadruple rule a A | B A X a A | B B X b A A b B B ⇒ X = A ⇒ X = B c c d d 20 / 26

Drawings of K 3 , 2 1 2 1 2 1 2 B 2 B 3 1 2 1 2 1 2 1 2 1 2 1 2 B 1 1 2 1 2 1 2 W 2 W 3 2 2 1 1 2 1 1 1 W 1 1 2 2 2 21 / 26

Drawings of K 3 , 2 : projections B 1 B 2 B 3 W 1 W 2 W 3 T 1 B A A A N N T 2 A B A N A N T 3 A A B N N A 22 / 26

Consistency for arbitrary k Theorem Consistency on subdrawings of K 2 , 3 (triples), K 2 , 4 (quadruples), and K 3 , 2 is sufficient for the existence of outer drawings of K k , n . Corollary Outer realizability of complete bipartite AT-graphs is in P 23 / 26

Proof steps k = 2 and arbitrary rotation system k = 3 and arbitrary rotation system : case analysis Generalize from k = 3 to arbitrary k 24 / 26

Other results Rotation systems of extendable (aka pseudolinear ) outer drawings ↔ suballowable sequences (Asinowski 2008) 25 / 26

Thank you! arXiv:1608.08324 To appear in Journal of Computational Geometry (JoCG) 26 / 26