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.

Hanani-Tutte for Radial Planarity

joint work with Radoslav Fulek Michael Pelsmajer

Marcus Schaefer

“We interpret this sentence as a philosophical view and not a mathematical claim.” Székely “We are taking the view that crossings

• f adjacent edges are trivial, and easily

got rid of.” Tutte

Adjacent edges do not cross in drawing minimizing number of crossings → graph drawable without non-adjacent crossings is planar. Obvious:

“We interpret this sentence as a philosophical view and not a mathematical claim.” Székely “We are taking the view that crossings

• f adjacent edges are trivial, and easily

got rid of.” Tutte

graph drawable without non-adjacent crossings is planar. Obvious ?

“We interpret this sentence as a philosophical view and not a mathematical claim.” Székely “We are taking the view that crossings

• f adjacent edges are trivial, and easily

got rid of.” Tutte

Hanani-Tutte Theorem (H‘34, T‘70)

(Weak) If graph has drawing without

• dd pairs, then graph is planar.

(Strong) If graph has drawing without independent odd pairs, then graph is planar.

pair of edges is even ↔ cross even # of times

• dd ↔ cross odd # of times

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

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

Radial Planarity

Radial HT Theorem

(Weak) If graph has radial drawing without

• dd 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

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

linear time recognition and embedding algorithm (PQ- trees), Bachmaier, Brandenburg, Forster, 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:

 𝑤𝑗+1 has lower edge  it doesn’t

Case: 𝑤𝑗+1 has lower edge e

Case: 𝑤𝑗+1 has no lower edge

𝑤𝑗+1 𝑄

Algorithm

G radial planar with rotation 𝜍 ↔ there is a radial planar drawing 𝐸 of 𝐻 with rotation 𝜍 ↔ given any radial drawing 𝐸 of 𝐻 with rotation 𝜍 there is a set of (𝑓, 𝑤)-moves and a set of 𝑓-twists in the resulting drawing 𝐸’ 𝑗𝐸′ 𝑓, 𝑔 ≡ 0 (𝑛𝑝𝑒 2) for all pairs (𝑓, 𝑔) of edges in 𝐻 given any radial drawing 𝐸 of 𝐻 with rotation 𝜍 there are x𝑓,v ∈ 0,1 , 𝑦𝑓 ∈ 0,1 , for all 𝑓 ∈ 𝐹, 𝑤 ∈ 𝑊, so that 𝑗𝐸 𝑓, 𝑔 + 𝑦𝑓,𝑔

1 + 𝑦𝑓,𝑔 2 + 𝑦𝑔,𝑓1 + 𝑦𝑔,𝑓2 + 𝑦𝑓 ≡ 0 (𝑛𝑝𝑒 2)

for all pairs (𝑓, 𝑔) of edges in 𝐻 , where 𝑦𝑓,𝑤 = 0 if 𝑓 does not overlap 𝑤 ↔ Polynomial time radial planarity algorithm, 𝑃(𝑜6) for given rotation system 𝑓 𝑤 (𝑓, 𝑤)-move 𝑓-twist

Radial HT Theorem

(Strong) If graph has radial drawing without

• dd independent pairs, then graph has equivalent

radial embedding.

Corollary

Radial level planarity can be tested using HT.

Actually

So

G radial planar ↔ there is a radial planar drawing 𝐸 of 𝐻 ↔ given any radial drawing 𝐸 of 𝐻 there is a set of (𝑓, 𝑤)-moves and a set of 𝑓-twists in the resulting drawing 𝐸’ 𝑗𝐸′ 𝑓, 𝑔 ≡ 0 (𝑛𝑝𝑒 2) for all independent pairs (𝑓, 𝑔) of edges in 𝐻 given any radial drawing 𝐸 of 𝐻 there are x𝑓,v ∈ 0,1 , 𝑦𝑓 ∈ 0,1 , for all 𝑓 ∈ 𝐹, 𝑤 ∈ 𝑊, so that 𝑗𝐸 𝑓, 𝑔 + 𝑦𝑓,𝑔

1 + 𝑦𝑓,𝑔 2 + 𝑦𝑔,𝑓1 + 𝑦𝑔,𝑓2 + 𝑦𝑓 ≡ 0 (𝑛𝑝𝑒 2)

for all independent pairs (𝑓, 𝑔) of edges in 𝐻, where 𝑦𝑓,𝑤 = 0 if 𝑓 does not overlap 𝑤 ↔ Polynomial time radial planarity algorithm, 𝑃(𝑜6) 𝑓 𝑤 (𝑓, 𝑤)-move 𝑓-twist

Hanani-Tutte theorems known for

• partially 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 (G1,G2) (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