graph drawing 2019

play

Graph Drawing 2019 Pr uhonice, September 17-19 On the 2-Colored - PowerPoint PPT Presentation

Jan 29, 2024 •375 likes •882 views

Graph Drawing 2019 Pr uhonice, September 17-19 On the 2-Colored Crossing Number Oswin Aichholzer 1 , Ruy Fabila-Monroy 2 , Adrian Fuchs 1 , Carlos Hidalgo-Toscano 2 , Irene Parada 1 , Birgit Vogtenhuber 1 , and Francisco Zaragoza 3 1 Graz

Graph Drawing 2019 Pr˚ uhonice, September 17-19 On the 2-Colored Crossing Number Oswin Aichholzer 1 , Ruy Fabila-Monroy 2 , Adrian Fuchs 1 , Carlos Hidalgo-Toscano 2 , Irene Parada 1 , Birgit Vogtenhuber 1 , and Francisco Zaragoza 3 1 Graz University of Technology, Austria 2 Cinvestav, Mexico 3 Universidad Aut´ onoma Metropolitana, Mexico
Rectilinear 2 -Colored Crossing Number Rectilinear Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • cr( D ) := number of crossings in ( D, χ ) • cr( G ) := min D cr( D ) 2
Rectilinear 2 -Colored Crossing Number Rectilinear Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • cr( D ) := number of crossings in ( D, χ ) • cr( G ) := min D cr( D ) 2
Rectilinear 2 -Colored Crossing Number Rectilinear Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • cr( D ) := number of crossings in ( D, χ ) • cr( G ) := min D cr( D ) 2
Rectilinear 2 -Colored Crossing Number Rectilinear Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • cr( D ) := number of crossings in ( D, χ ) • cr( G ) := min D cr( D ) 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Rectilinear 2 -Colored Crossing Number Given: (straight-line drawing D of) graph G = ( V, E ) . • 2-edge-coloring χ of G : one of 2 colors per edge • cr 2 ( D, χ ) := number of monochromatic crossings in ( D, χ ) • cr 2 ( D ) := min χ cr 2 ( D, χ ) • cr 2 ( G ) := min D cr 2 ( D ) • Determining cr 2 ( G ) and even cr 2 ( D ) is NP-hard • Goal: find bounds on cr 2 ( G ) and cr 2 ( D ) for G = K n . 2
Main Results • Lower and upper bounds on cr 2 ( K n ) : � n � � n � 1 + Θ( n 3 ) < cr 2 ( K n ) < 0 . 11798016 + Θ( n 3 ) 33 4 4 • Ratio between cr 2 ( K n ) and cr( K n ) : cr 2 ( K n ) lim cr( K n ) < 0 . 31049652 n →∞ • Ratio for any fixed straight-line drawing D of K n with sufficiently large n : cr 2 ( D ) cr( D ) < 1 2 − c for some const. c > 0 3
Main Results • Lower and upper bounds on cr 2 ( K n ) : � n � � n � 1 + Θ( n 3 ) < cr 2 ( K n ) < 0 . 11798016 + Θ( n 3 ) 33 4 4 • Ratio between cr 2 ( K n ) and cr( K n ) : cr 2 ( K n ) lim cr( K n ) < 0 . 31049652 n →∞ • Ratio for any fixed straight-line drawing D of K n with sufficiently large n : cr 2 ( D ) cr( D ) < 1 2 − c for some const. c > 0 3
Main Results • Lower and upper bounds on cr 2 ( K n ) : � n � � n � 1 + Θ( n 3 ) < cr 2 ( K n ) < 0 . 11798016 + Θ( n 3 ) 33 4 4 • Ratio between cr 2 ( K n ) and cr( K n ) : cr 2 ( K n ) lim cr( K n ) < 0 . 31049652 n →∞ • Ratio for any fixed straight-line drawing D of K n with sufficiently large n : cr 2 ( D ) cr( D ) < 1 2 − c for some const. c > 0 3
Duplication Process → drawing D ′ of K 2 m • Duplication: drawing D of K m − cr 2 of D ′ : independent of colors for small edges! • Best matching edges: half of the edges of each color on each side ⇐ in general not possible! • ”Nice” matching edges: ◮ halve the larger color class at the point ◮ split the smaller color class as good as possible 4
Duplication Process → drawing D ′ of K 2 m • Duplication: drawing D of K m − cr 2 of D ′ : independent of colors for small edges! • Best matching edges: half of the edges of each color on each side ⇐ in general not possible! • ”Nice” matching edges: ◮ halve the larger color class at the point ◮ split the smaller color class as good as possible 4
Duplication Process → drawing D ′ of K 2 m • Duplication: drawing D of K m − cr 2 of D ′ : independent of colors for small edges! • Best matching edges: half of the edges of each color on each side ⇐ in general not possible! • ”Nice” matching edges: ◮ halve the larger color class at the point ◮ split the smaller color class as good as possible 4
Duplication Process → drawing D ′ of K 2 m • Duplication: drawing D of K m − cr 2 of D ′ : independent of colors for small edges! • Best matching edges: half of the edges of each color on each side ⇐ in general not possible! • ”Nice” matching edges: ◮ halve the larger color class at the point ◮ split the smaller color class as good as possible 4
Duplication Process → drawing D ′ of K 2 m • Duplication: drawing D of K m − cr 2 of D ′ : independent of colors for small edges! • Best matching edges: half of the edges of each color on each side ⇐ in general not possible! • ”Nice” matching edges: ◮ halve the larger color class at the point ◮ split the smaller color class as good as possible per original crossing: 16 crossings 4
Duplication Process → drawing D ′ of K 2 m • Duplication: drawing D of K m − cr 2 of D ′ : independent of colors for small edges! • Best matching edges: half of the edges of each color on each side ⇐ in general not possible! • ”Nice” matching edges: ◮ halve the larger color class at the point ◮ split the smaller color class as good as possible per original edge: 1 crossing 4
Duplication Process → drawing D ′ of K 2 m • Duplication: drawing D of K m − cr 2 of D ′ : independent of colors for small edges! • Best matching edges: half of the edges of each color on each side ⇐ in general not possible! p p 1 p 2 • ”Nice” matching edges: ◮ halve the larger color class at the point ◮ split the smaller color class as good as possible except for matching edges 4

Download Presentation

Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

More recommend