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

On the 2-Colored Crossing Number

Oswin Aichholzer1, Ruy Fabila-Monroy2, Adrian Fuchs1, Carlos Hidalgo-Toscano2, Irene Parada1, Birgit Vogtenhuber1, and Francisco Zaragoza3

1 Graz University of Technology, Austria 2 Cinvestav, Mexico 3 Universidad Aut´

• noma Metropolitana, Mexico

Graph Drawing 2019

Pr˚ uhonice, September 17-19

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• cr(D) := number of

crossings in (D, χ)

• cr(G) := minD cr(D)

Rectilinear Crossing Number

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• cr(D) := number of

crossings in (D, χ)

• cr(G) := minD cr(D)

Rectilinear Crossing Number

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• cr(D) := number of

crossings in (D, χ)

• cr(G) := minD cr(D)

Rectilinear Crossing Number

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• cr(D) := number of

crossings in (D, χ)

• cr(G) := minD cr(D)

Rectilinear Crossing Number

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Rectilinear 2-Colored Crossing Number

Given: (straight-line drawing D of) graph G = (V, E).

• 2-edge-coloring χ of G:
• ne of 2 colors per edge
• cr2(D, χ) := number of

monochromatic crossings in (D, χ)

• cr2(D) := minχ cr2(D, χ)
• cr2(G) := minD cr2(D)
• Determining cr2(G) and

even cr2(D) is NP-hard

• Goal: find bounds on cr2(G) and cr2(D) for G = Kn.

2

Main Results

• Lower and upper bounds on cr2(Kn):

1 33 n 4

• + Θ(n3) < cr2(Kn) < 0.11798016

n 4

• + Θ(n3)
• Ratio between cr2(Kn) and cr(Kn):

lim

n→∞

cr2(Kn) cr(Kn) < 0.31049652

• Ratio for any fixed straight-line drawing D of Kn

with sufficiently large n: cr2(D) cr(D) < 1 2 − c for some const. c > 0

3

Main Results

• Lower and upper bounds on cr2(Kn):

1 33 n 4

• + Θ(n3) < cr2(Kn) < 0.11798016

n 4

• + Θ(n3)
• Ratio between cr2(Kn) and cr(Kn):

lim

n→∞

cr2(Kn) cr(Kn) < 0.31049652

• Ratio for any fixed straight-line drawing D of Kn

with sufficiently large n: cr2(D) cr(D) < 1 2 − c for some const. c > 0

3

Main Results

• Lower and upper bounds on cr2(Kn):

1 33 n 4

• + Θ(n3) < cr2(Kn) < 0.11798016

n 4

• + Θ(n3)
• Ratio between cr2(Kn) and cr(Kn):

lim

n→∞

cr2(Kn) cr(Kn) < 0.31049652

• Ratio for any fixed straight-line drawing D of Kn

with sufficiently large n: cr2(D) cr(D) < 1 2 − c for some const. c > 0

3

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2

except for matching edges

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2

incident edges: 2 additional crossings with matching edge

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2

incident edge pairs: 4 additional crossings

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2

• pposite incident edge pairs: no additional crossings

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2

small edges: no crossings

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2 L(p) R(p) cr(D′) = 16cr(D) +

• n

2

• −n +
• p∈D
• 4
• L(p)

2

• + 4
• R(p)

2

• + 2 (L(p)+R(p))
• crossings in D

edges in D

• edgepairs incident to p ∈ D

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2 L(p) R(p) cr(D′) = 16cr(D) +

• n

2

• −n +
• p∈D
• 4
• L(p)

2

• + 4
• R(p)

2

• + 2 (L(p)+R(p))
• crossings in D

edges in D

• edgepairs incident to p ∈ D
• Best matching edges: half of the edges on each side

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2 Bl(p) Br(p) Rl(p) Rr(p)

q

q1 q2

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2 Bl(p) Br(p) Rl(p) Rr(p) cr2(D′, χ′) = 16cr2(D, χ) +

• n

2

• − n

+

• p∈D

4

• Bl(p)

2

• +
• Rl(p)

2

• +
• Br(p)

2

• +
• Rr(p)

2

• +
• p∈D

2 (Hl(p)+Hr(p)) ⇐ Hi ∈ {Bi, Ri}

q

q1 q2

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

p

p1 p2 Bl(p) Br(p) Rl(p) Rr(p) cr2(D′, χ′) = 16cr2(D, χ) +

• n

2

• − n

+

• p∈D

4

• Bl(p)

2

• +
• Rl(p)

2

• +
• Br(p)

2

• +
• Rr(p)

2

• +
• p∈D

2 (Hl(p)+Hr(p)) ⇐ Hi ∈ {Bi, Ri}

q

q1 q2

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

cr2(D′, χ′) = 16cr2(D, χ) +

• n

2

• − n

+

• p∈D

4

• Bl(p)

2

• +
• Rl(p)

2

• +
• Br(p)

2

• +
• Rr(p)

2

• +
• p∈D

2 (Hl(p)+Hr(p)) ⇐ Hi ∈ {Bi, Ri}

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

cr2(D′, χ′) = 16cr2(D, χ) +

• n

2

• − n

+

• p∈D

4

• Bl(p)

2

• +
• Rl(p)

2

• +
• Br(p)

2

• +
• Rr(p)

2

• +
• p∈D

2 (Hl(p)+Hr(p)) ⇐ Hi ∈ {Bi, Ri}

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

cr2(D′, χ′) = 16cr2(D, χ) +

• n

2

• − n

+

• p∈D

4

• Bl(p)

2

• +
• Rl(p)

2

• +
• Br(p)

2

• +
• Rr(p)

2

• +
• p∈D

2 (Hl(p)+Hr(p)) ⇐ Hi ∈ {Bi, Ri}

4

Duplication Process

• Duplication: drawing D of Km −

→ drawing D′ of K2m cr2 of D′: independent of colors for small edges!

• Best matching edges: half of the edges of each color
• n 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

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

p

p1 p2 Bl(p) Br(p) Rl(p) Rr(p)

q

q1 q2

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

p

p1 p2 Bl(p) Br(p) Rl(p) Rr(p)

q

q1 q2

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

p

p1 p2 Bl(p) Br(p) Rl(p) Rr(p)

q

q1 q2

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

p

p1 p2 Bl(p) Br(p) Rl(p) Rr(p)

q

q1 q2

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

5

Upper Bound for cr2(Kn)

• Duplication: drawing D of Km → drawing D′ of K2m
• Matching for D′: for each p ∈ D, independently choose

matching edges for p1, p2 and the color of p1p2 choice depends on: |Ri(p)|, |Bi(p)|, i ∈ {l, r}, color of pq

◮ several cases, choices with good recursive behavior

• Repeated duplication: D → drawing Dk of K2km

◮ involved analysis + exact counting yields

cr2(Dk, χk) = 24A

m4

n

4

• + Θ(n3)

n = m 2k

(A: constant depending on D, χ, and the matching M)

• Plugging in a good initial (D, χ, M) gives

cr2(Kn) < 0.11798016 n

4

• +Θ(n3)

5

Main Results

• Lower and upper bounds on cr2(Kn):

1 33 n 4

• + Θ(n3) < cr2(Kn) < 0.11798016

n 4

• + Θ(n3)
• Ratio between cr2(Kn) and cr(Kn):

lim

n→∞

cr2(Kn) cr(Kn) < 0.31049652

• Ratio for any fixed straight-line drawing D of Kn

with sufficiently large n: cr2(D) cr(D) < 1 2 − c for some c > 0

6

Open Problems

• What is the computational complexity of determinging

cr2(D) for a given straight-line drawing D of Kn? ⇔ How fast can we solve max-cut on the segment intersection graph induced by D?

• What can we say about the structure of point sets that

minimize cr2(Kn)?

• Is it true that the maximum for cr2(D)/cr(D) is

uniquely obtained for point sets in convex position?

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Open Problems

• What is the computational complexity of determinging

cr2(D) for a given straight-line drawing D of Kn? ⇔ How fast can we solve max-cut on the segment intersection graph induced by D?

• What can we say about the structure of point sets that

minimize cr2(Kn)?

• Is it true that the maximum for cr2(D)/cr(D) is

uniquely obtained for point sets in convex position?

Thank you for your attention!

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