A Method for Depicting Social Relationships Obtained by Sociometric - PowerPoint PPT Presentation

A Method for Depicting Social Relationships Obtained by Sociometric Testing, Mary L. Northway. Sociometry , 3(2), 1940.

“We are taking the view that crossings “We interpret this sentence as a of adjacent edges are trivial, and easily philosophical view and not a got rid of.” Tutte mathematical claim.” Székely Hanani-Tutte for Radial Planarity Marcus Schaefer joint work with Radoslav Fulek Michael Pelsmajer

“We are taking the view that crossings “We interpret this sentence as a of adjacent edges are trivial, and easily philosophical view and not a got rid of.” Tutte mathematical claim.” Székely Obvious: Adjacent edges do not cross in drawing minimizing number of crossings → graph drawable without non-adjacent crossings is planar.

“We are taking the view that crossings “We interpret this sentence as a of adjacent edges are trivial, and easily philosophical view and not a got rid of.” Tutte mathematical claim.” Székely Obvious ? graph drawable without non-adjacent crossings is planar.

pair of edges is even ↔ cross even # of times odd ↔ cross odd # of times Hanani-Tutte Theorem (H‘34, T‘70) (Weak) If graph has drawing without odd pairs , then graph is planar. (Strong) If graph has drawing without independent odd pairs , then graph is planar.

monotone drawings: edges as functions

Monotone HT-Theorems (Weak) Monotone drawing without odd pairs , then equivalent monotone embedding. [Pach, Tóth , ’04, ’11; Fulek , Pelsmajer, Schaefer, Štefankovič (Strong) Monotone drawing without independent odd pairs, then monotone embedding. [Fulek, Pelsmajer, Schaefer, Štefankovič, ‘11 ] equivalent : rotation system remains same

Radial Planarity A Method for Depicting Social Relationships Obtained by Sociometric Testing, Mary L. Northway. Sociometry , 3(2), 1940.

Radial HT Theorem (Weak) If graph has radial drawing without odd pairs , then graph has equivalent radial embedding. Corollary Radial level planarity can be tested using HT if rotation is given.

István Orosz. http://twistedsifter.com/2015/01/anamorphic-art-by-istvan-orosz /

Radial Level-Planarity linear time recognition and embedding algorithm (PQ- trees), Bachmaier, Brandenburg, Forster, 2004. from Bachmaier, Brandenburg, Forster. Radial Level Planarity Testing and Embedding in Linear Time , 2004

Radial Level-Planarity to Radial Planarity from Bachmaier, Brandenburg, Forster. Radial Level Planarity Testing and Embedding in Linear Time , 2004

Proof: can assume:  at most one vertex per level  graph is connected  (potential) faces are known  faces have at most two local maxima and two local minima  any (single) edge can be made crossing-free fix edges below each level; two cases:  𝑤 𝑗+1 has lower edge  it doesn’t

Case: 𝑤 𝑗+1 has lower edge e

Case: 𝑤 𝑗+1 has no lower edge 𝑄 𝑤 𝑗+1

Algorithm G radial planar with rotation 𝜍 𝑓 -twist ↔ there is a radial planar drawing 𝐸 of 𝐻 with rotation 𝜍 ↔ given any radial drawing 𝐸 of 𝐻 with rotation 𝜍 𝑓 there is a set of (𝑓, 𝑤) -moves and a set of 𝑓 -twists in the resulting drawing 𝐸’ 𝑤 𝑗 𝐸′ 𝑓, 𝑔 ≡ 0 (𝑛𝑝𝑒 2) for all pairs (𝑓, 𝑔) of edges in 𝐻 (𝑓, 𝑤) -move ↔ given any radial drawing 𝐸 of 𝐻 with rotation 𝜍 there are x 𝑓,v ∈ 0,1 , 𝑦 𝑓 ∈ 0,1 , for all 𝑓 ∈ 𝐹, 𝑤 ∈ 𝑊 , so that 𝑗 𝐸 𝑓, 𝑔 + 𝑦 𝑓,𝑔 1 + 𝑦 𝑓,𝑔 2 + 𝑦 𝑔,𝑓 1 + 𝑦 𝑔,𝑓 2 + 𝑦 𝑓 ≡ 0 (𝑛𝑝𝑒 2) for all pairs (𝑓, 𝑔) of edges in 𝐻 , where 𝑦 𝑓,𝑤 = 0 if 𝑓 does not overlap 𝑤 Polynomial time radial planarity algorithm, 𝑃(𝑜 6 ) for given rotation system

Actually Radial HT Theorem (Strong) If graph has radial drawing without odd independent pairs , then graph has equivalent radial embedding. Corollary Radial level planarity can be tested using HT.

So G radial planar ↔ there is a radial planar drawing 𝐸 of 𝐻 𝑓 -twist ↔ given any radial drawing 𝐸 of 𝐻 there is a set of (𝑓, 𝑤) -moves and a set of 𝑓 -twists 𝑓 in the resulting drawing 𝐸’ 𝑗 𝐸′ 𝑓, 𝑔 ≡ 0 (𝑛𝑝𝑒 2) 𝑤 for all independent pairs (𝑓, 𝑔) of edges in 𝐻 (𝑓, 𝑤) -move ↔ given any radial drawing 𝐸 of 𝐻 there are x 𝑓,v ∈ 0,1 , 𝑦 𝑓 ∈ 0,1 , for all 𝑓 ∈ 𝐹, 𝑤 ∈ 𝑊 , so that 𝑗 𝐸 𝑓, 𝑔 + 𝑦 𝑓,𝑔 1 + 𝑦 𝑓,𝑔 2 + 𝑦 𝑔,𝑓 1 + 𝑦 𝑔,𝑓 2 + 𝑦 𝑓 ≡ 0 (𝑛𝑝𝑒 2) for all independent pairs (𝑓, 𝑔) of edges in 𝐻 , where 𝑦 𝑓,𝑤 = 0 if 𝑓 does not overlap 𝑤 Polynomial time radial planarity algorithm, 𝑃(𝑜 6 )

Hanani-Tutte theorems known for • p artially embedded planarity (S ‘14) • partial rotation (with or without flips) • partial planarity (S ‘15) • x-monotone (Fulek, Pelsmajer, S, Štefankovič, ‘11) • level- planarity (implicit in FPSS, ‘11) • c-planarity for 2 clusters (Fulek, Kyncl, Malinovič , Pālvölgyi ‘14) • special cases of simultaneous planarity of (G 1 ,G 2 ) (S’14) • projective planarity (PSS, ’09) There are counterexamples for • c-planarity for 3 clusters (Fulek, Kyncl, Malinovič , Pālvölgyi ’14) • simultaneous planarity (Gutwenger , Mutzel, S ’14, based 3 -cluster) Open • strip planarity (weak case known: Fulek ’14) • toroidality