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k-colored Point-set Embeddability of Graphs
Peter Korcsok and Michael Skotnicamentor Periklis Papakonstantinou
Charles University, Prague
DIMACS REU 2018Rutgers University, June 4, 2018
1 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Point-set Embedding (PSE)
DefinitionWe are given a planar graph G = (V,E) and a point set S(|V | = |S|).
A point-set embedding of G on S is a planar drawing such thateach vertex is represented by a distinct point of S.
A curve complexity (CC) of the PSE is the maximum number ofbends along any edge.
5 6
4
12
3
7
G S
2 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Point-set Embedding (PSE)
DefinitionWe are given a planar graph G = (V,E) and a point set S(|V | = |S|).
A point-set embedding of G on S is a planar drawing such thateach vertex is represented by a distinct point of S.
A curve complexity (CC) of the PSE is the maximum number ofbends along any edge.
5 6
4
12
3
7
G S
1
7
3
25
4
6
2 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Point-set Embedding (PSE)
DefinitionWe are given a planar graph G = (V,E) and a point set S(|V | = |S|).
A point-set embedding of G on S is a planar drawing such thateach vertex is represented by a distinct point of S.
A curve complexity (CC) of the PSE is the maximum number ofbends along any edge.
5 6
4
12
3
7
G S
1
7
3
25
4
6
CC = 1
2 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Point-set Embedding (PSE)
DefinitionWe are given a planar graph G = (V,E) and a point set S(|V | = |S|).
A point-set embedding of G on S is a planar drawing such thateach vertex is represented by a distinct point of S.
A curve complexity (CC) of the PSE is the maximum number ofbends along any edge.
5 6
4
12
3
7
G S
71
2
36
5
4
CC = 2
2 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
More Colors and More Graphs
DefinitionIn a colored PSE, both vertices and points are colored – a vertexcan be represented only by a point of the same color.
G S
5 6
4
12
3
7
ProblemGiven a family of planar graphs G and a number of colors k, wewant to know the worst-case CC for any graph G ∈ G and anypoint set S.
3 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
More Colors and More Graphs
DefinitionIn a colored PSE, both vertices and points are colored – a vertexcan be represented only by a point of the same color.
G S
5 6
4
12
3
7
14
5
6
7
3
2
ProblemGiven a family of planar graphs G and a number of colors k, wewant to know the worst-case CC for any graph G ∈ G and anypoint set S.
3 / 7k-colored Point-set Embeddability of Graphs
![Page 8: Charles University, Prague - DIMACSreu.dimacs.rutgers.edu/~michaels/Presentation1.pdf · Introduction Motivation A Little Bit of History Our Plan Known Results PSE: upperand lowerbounds](https://reader030.vdocuments.us/reader030/viewer/2022041105/5f066d897e708231d417f19f/html5/thumbnails/8.jpg)
Introduction Motivation A Little Bit of History Our Plan
More Colors and More Graphs
DefinitionIn a colored PSE, both vertices and points are colored – a vertexcan be represented only by a point of the same color.
G S
5 6
4
12
3
7
14
5
6
7
3
2
ProblemGiven a family of planar graphs G and a number of colors k, wewant to know the worst-case CC for any graph G ∈ G and anypoint set S.
3 / 7k-colored Point-set Embeddability of Graphs
![Page 9: Charles University, Prague - DIMACSreu.dimacs.rutgers.edu/~michaels/Presentation1.pdf · Introduction Motivation A Little Bit of History Our Plan Known Results PSE: upperand lowerbounds](https://reader030.vdocuments.us/reader030/viewer/2022041105/5f066d897e708231d417f19f/html5/thumbnails/9.jpg)
Introduction Motivation A Little Bit of History Our Plan
Flow map drawing
k = 2 colors
G is a rooted tree (root has its unique color and it representsColorado), S is a map of the United States
Migration from Colorado[Verbeek, Buchin, Speckmann IEEE TVCG 2011]
4 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Flow map drawing
k = 2 colors
G is a rooted tree (root has its unique color and it representsColorado), S is a map of the United States
Migration from Colorado[Verbeek, Buchin, Speckmann IEEE TVCG 2011]
4 / 7k-colored Point-set Embeddability of Graphs
![Page 11: Charles University, Prague - DIMACSreu.dimacs.rutgers.edu/~michaels/Presentation1.pdf · Introduction Motivation A Little Bit of History Our Plan Known Results PSE: upperand lowerbounds](https://reader030.vdocuments.us/reader030/viewer/2022041105/5f066d897e708231d417f19f/html5/thumbnails/11.jpg)
Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1
2
3
· · · · · · · · · · · · · · · · · ·
n
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial]
2
3
· · · · · · · · · · · · · · · · · ·
n
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 [2]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial]
2
3
· · · · · · · · · · · · · · · · · ·
n
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial]
2
3
· · · · · · · · · · · · · · · · · ·
n
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2
3
· · · · · · · · · · · · · · · · · ·
n
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2
3
· · · · · · · · · · · · · · · · · ·
n O(n) [9]Ω(n) [9]
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2
3
· · · · · · · · · · · · · · · · · ·
n O(n) O(n) O(n) O(n) O(n) [9]Ω(n) [9] Ω(n) Ω(n) Ω(n) Ω(n)
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2 1 [5] 1 [6] 5 [3]1 [7]
3
· · · · · · · · · · · · · · · · · ·
n O(n) O(n) O(n) O(n) O(n) [9]Ω(n) [9] Ω(n) Ω(n) Ω(n) Ω(n)
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2 1 [5] 1 [6] 5 5 [3]1 [7] 1 1 1
3
· · · · · · · · · · · · · · · · · ·
n O(n) O(n) O(n) O(n) O(n) [9]Ω(n) [9] Ω(n) Ω(n) Ω(n) Ω(n)
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
![Page 20: Charles University, Prague - DIMACSreu.dimacs.rutgers.edu/~michaels/Presentation1.pdf · Introduction Motivation A Little Bit of History Our Plan Known Results PSE: upperand lowerbounds](https://reader030.vdocuments.us/reader030/viewer/2022041105/5f066d897e708231d417f19f/html5/thumbnails/20.jpg)
Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2 1 [5] 1 [6] 5 5 [3] O(n)1 [7] 1 1 1 Ω(n) [1]
3
· · · · · · · · · · · · · · · · · ·
n O(n) O(n) O(n) O(n) O(n) [9]Ω(n) [9] Ω(n) Ω(n) Ω(n) Ω(n)
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
![Page 21: Charles University, Prague - DIMACSreu.dimacs.rutgers.edu/~michaels/Presentation1.pdf · Introduction Motivation A Little Bit of History Our Plan Known Results PSE: upperand lowerbounds](https://reader030.vdocuments.us/reader030/viewer/2022041105/5f066d897e708231d417f19f/html5/thumbnails/21.jpg)
Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2 1 [5] 1 [6] 5 5 [3] O(n)1 [7] 1 1 1 Ω(n) [1]
3 5 [4]Ω( 3√n) [4] Ω( 3
√n) [3]
· · · · · · · · · · · · · · · · · ·
n O(n) O(n) O(n) O(n) O(n) [9]Ω(n) [9] Ω(n) Ω(n) Ω(n) Ω(n)
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Known Results
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2 1 [5] 1 [6] 5 5 [3] O(n)1 [7] 1 1 1 Ω(n) [1]
3 5 [4] O(n) O(n) O(n) O(n)1 Ω( 3
√n) [4] Ω( 3
√n) Ω( 3
√n) [3] Ω(n)
· · · · · · · · · · · · · · · · · ·
n O(n) O(n) O(n) O(n) O(n) [9]Ω(n) [9] Ω(n) Ω(n) Ω(n) Ω(n)
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
5 / 7k-colored Point-set Embeddability of Graphs
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Introduction Motivation A Little Bit of History Our Plan
Our Plan
PSE: upper and lower bounds on CC
paths caterp. trees outerpl. planar
1 0 0 0 0 [2] 2 [8]0 [trivial] 0 [trivial] 0 [trivial] 0 [trivial] 2 [8]
2 1 [5] 1 [6] 5 5 [3] O(n)1 [7] 1 1 1 Ω(n) [1]
3 5 [4] O(n) O(n) O(n) O(n)1 Ω( 3
√n) [4] Ω( 3
√n) Ω( 3
√n) [3] Ω(n)
· · · · · · · · · · · · · · · · · ·
n O(n) O(n) O(n) O(n) O(n) [9]Ω(n) [9] Ω(n) Ω(n) Ω(n) Ω(n)
[1] Badent, Di Giacomo, Liotta TCS 2008; [2] Bose CGTA 2002; [3] Di Giacomo et al. JGAA 2008;[4] Di Giacomo et al. GD 2017; [5] Di Giacomo, Liotta, Trotta IJFCS 2006; [6] Hančl pers.comm.;[7] Kaneko, Kano, Suzuki TTGG 2004; [8] Kaufmann, Wiese JGAA 2002; [9] Pach, Wenger G&C 2001
6 / 7k-colored Point-set Embeddability of Graphs
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Thank you!
7 / 7k-colored Point-set Embeddability of Graphs