[PPT] - Extending Simple Drawings Alan Arroyo 1 , Martin Derka 2 , and Irene PowerPoint Presentation

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Irene Parada Extending Simple Drawings

Extending Simple Drawings

Alan Arroyo1, Martin Derka2, and Irene Parada3

1 IST Austria 2 Carleton University, Canada 3 Graz University of Technology, Austria

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Irene Parada Extending Simple Drawings

Simple drawings

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Irene Parada Extending Simple Drawings

Simple drawings

Not simple drawings:

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Irene Parada Extending Simple Drawings

Simple drawings

Locally fixed: now they are!

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Irene Parada Extending Simple Drawings

Simple drawings

Locally fixed: now they are! Drawings that minimize the total number of crossings are simple.

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Irene Parada Extending Simple Drawings

Extending a partial representation

Abstract graph G (Partial) representation of a subgraph of G

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Irene Parada Extending Simple Drawings

Extending a partial representation

Abstract graph G (Partial) representation of a subgraph of G

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Irene Parada Extending Simple Drawings

Extending a partial representation

Abstract graph G (Partial) representation of a subgraph of G

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Irene Parada Extending Simple Drawings

Extending a partial representation

[Bagheri, Razzazi ’10]
[Jel´

ınek, Kratochv´ ıl, Rutter ’13]

[Angelini et. al. ’15]
[Mchedlidze, N¨
llenburg,

Rutter ’15]

[Br¨

uckner, Rutter ’17]

[Da Lozzo, Di Battista,

Frati ’19]

[Patrignani ’06]
[Klav´

ık, Kratochv´ ıl, Krawczyk, Walczak ’12]

[Chaplick et. al. ’14]
[Klav´

ık, Kratochv´ ıl, Otachi, Saitoh ’15]

[Klav´

ık et. al. ’17]

[Klav´

ık et. al. ’17]

[Chaplick et. al. ’18]
[Chaplick, Fulek, Klav´

ık ’19] Extending partial drawings

f planar graphs:

Extending partial rep. that are not drawings:

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

Given a simple drawing D(G) of a graph G = (V, E) we want to insert a set of edges (of the complement of G) s.t. the result is a simple drawing with D(G) as a subdrawing.

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

In straight-line drawings trivially YES

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

In pseudolinear drawings YES by Levis enlargement lemma

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v [Kynˇ cl ’13] uv cannot be added

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v [Kynˇ cl ’13] uv cannot be added

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v [Kynˇ cl ’13] u v ... ... uv cannot be added Km,n

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v [Kynˇ cl ’13] u v ... ... uv cannot be added u v ... Km,n Kn \ uv

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v [Kynˇ cl ’13] u v ... ... uv cannot be added u v ... Km,n Kn \ uv What about matchings?

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Irene Parada Extending Simple Drawings

Can we always insert the remaining edges?

u v [Kynˇ cl ’13] u v ... ... uv cannot be added u v ... Km,n Kn \ uv What about matchings? [Kynˇ cl, Pach, Radoiˇ ci´ c, T´

th ’14]

u v u v

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Irene Parada Extending Simple Drawings

Inserting a set of edges is NP-complete

Reduction from monotone 3SAT.

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Irene Parada Extending Simple Drawings

Inserting a set of edges is NP-complete

Reduction from monotone 3SAT.

Variable gadget u v

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Irene Parada Extending Simple Drawings

Inserting a set of edges is NP-complete

Reduction from monotone 3SAT.

Variable gadget u v

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Irene Parada Extending Simple Drawings

Inserting a set of edges is NP-complete

Reduction from monotone 3SAT.

Variable gadget u v v u Clause gadget

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Irene Parada Extending Simple Drawings

Inserting a set of edges is NP-complete

Reduction from monotone 3SAT.

Variable gadget u v v u Clause gadget Wire gadget u v

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Irene Parada Extending Simple Drawings

Inserting a set of edges is NP-complete

Reduction from monotone 3SAT.

true false Clause gadgets Wire gadgets Variable gadget

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Irene Parada Extending Simple Drawings

Finding the largest extension is APX-hard

Reduction from maximum indep. set in max. deg. ≤ 3.

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Irene Parada Extending Simple Drawings

Finding the largest extension is APX-hard

Reduction from maximum indep. set in max. deg. ≤ 3.

v u Vertex gadget

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Irene Parada Extending Simple Drawings

Finding the largest extension is APX-hard

Reduction from maximum indep. set in max. deg. ≤ 3.

v u Vertex gadget u v Edge gadget

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Irene Parada Extending Simple Drawings

Finding the largest extension is APX-hard

Reduction from maximum indep. set in max. deg. ≤ 3.

v u Vertex gadget u v Edge gadget

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Irene Parada Extending Simple Drawings

Finding the largest extension is APX-hard

Reduction from maximum indep. set in max. deg. ≤ 3.

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Irene Parada Extending Simple Drawings

Inserting one single edge

...

u v An edge may be added in exponentially many ways.

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Irene Parada Extending Simple Drawings

Inserting one single edge

...

u v An edge may be added in exponentially many ways. View in the dual: Heterochromatic path.

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Irene Parada Extending Simple Drawings

Inserting one single edge

...

u v An edge may be added in exponentially many ways. View in the dual: Heterochromatic path.

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Irene Parada Extending Simple Drawings

Inserting one single edge

...

u v An edge may be added in exponentially many ways. Theorem: If {u, v} is a dominating set for G then the problem of extending D(G) with the edge uv can be decided in polynomial time. View in the dual: Heterochromatic path.

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Irene Parada Extending Simple Drawings

Conclusions

Results:

Deciding if we can insert a set of k edges is NP-complete.
Maximizing the number of edges from a given set that we can

insert is APX-hard.

Under certain conditions we can decide in polynomial time if we

can insert a particular edge.

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Irene Parada Extending Simple Drawings

Conclusions

Results:

Deciding if we can insert a set of k edges is NP-complete.
Maximizing the number of edges from a given set that we can

insert is APX-hard.

Under certain conditions we can decide in polynomial time if we

can insert a particular edge. Question:

Computational complexity of deciding whether a given edge can

be inserted?

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Irene Parada Extending Simple Drawings

Conclusions

A. Arroyo, F. Klute, I. Parada, R. Seidel, B. Vogtenhuber, T. Wiedera.

Extending simple drawings with one edge is hard. arXiv:1909.07347. Results:

Deciding if we can insert a set of k edges is NP-complete.
Maximizing the number of edges from a given set that we can

insert is APX-hard.

Under certain conditions we can decide in polynomial time if we

can insert a particular edge. Question:

Computational complexity of deciding whether a given edge can

be inserted?

S

l

v e d

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Irene Parada Extending Simple Drawings

Conclusions

A. Arroyo, F. Klute, I. Parada, R. Seidel, B. Vogtenhuber, T. Wiedera.

Extending simple drawings with one edge is hard. arXiv:1909.07347. Results:

Deciding if we can insert a set of k edges is NP-complete.
Maximizing the number of edges from a given set that we can

insert is APX-hard.

Under certain conditions we can decide in polynomial time if we

can insert a particular edge.

Thank you!

Question:

Computational complexity of deciding whether a given edge can

be inserted?

S

l