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
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