SLIDE 1 Variants of the Segment Number of a Graph
Yoshio Okamoto
University of Electro-Communications, Ch¯
RIKEN Center for Advanced Intelligence Project, Tokyo, Japan
Alexander Ravsky
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Science of Ukraine, Lviv, Ukraine
Alexander Wolff
Julius-Maximilians-Universit¨ at W¨ urzburg, Germany
SLIDE 2
Measures of Visual Complexity
SLIDE 3
Measures of Visual Complexity
[Wade & Chu 1994] Slope number
SLIDE 4
Measures of Visual Complexity
[Schulz 2015] Arc number [Wade & Chu 1994] Slope number
SLIDE 5
Measures of Visual Complexity
[Schulz 2015] Arc number Segment number (seg2(G)) [Dujmovi´ c et al. 2007] [Wade & Chu 1994] Slope number
SLIDE 6
Measures of Visual Complexity
SLIDE 7
Measures of Visual Complexity
Line cover number [Chaplick et al. 2016]
SLIDE 8 Measures of Visual Complexity
Line cover number [Chaplick et al. 2016] ρ1
2(G)
[Scherm 2016]
SLIDE 9 Measures of Visual Complexity
Line cover number [Chaplick et al. 2016] ρ1
2(G)
ρ1
3(G)
[Scherm 2016]
SLIDE 10
Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G.
SLIDE 11
Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar.
SLIDE 12 Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar.
seg2(K4) = 6
SLIDE 13 Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar. seg∠(G), where G is planar, drawings are 2D, bends are OK, but no crossings.
seg2(K4) = 6
SLIDE 14 Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar. seg∠(G), where G is planar, drawings are 2D, bends are OK, but no crossings.
seg2(K4) = 6 seg∠(K4) = 5
SLIDE 15 Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar. seg3(G), where drawings are 3D, no bends, no crossings. seg∠(G), where G is planar, drawings are 2D, bends are OK, but no crossings.
seg2(K4) = 6 seg∠(K4) = 5
SLIDE 16 Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar. seg3(G), where drawings are 3D, no bends, no crossings. seg∠(G), where G is planar, drawings are 2D, bends are OK, but no crossings.
seg2(K4) = 6 seg∠(K4) = 5 seg3(K3,3) = 7
SLIDE 17 Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar. seg3(G), where drawings are 3D, no bends, no crossings. seg∠(G), where G is planar, drawings are 2D, bends are OK, but no crossings. seg×(G), where drawings are 2D, crossings are OK, but no bends and no overlaps.
seg2(K4) = 6 seg∠(K4) = 5 seg3(K3,3) = 7
SLIDE 18 Segment Number Variants of Graphs
The segment number of a graph G is the minimum number of segments constituting a straight-line drawing of G. seg2(G), where G is planar. seg3(G), where drawings are 3D, no bends, no crossings. seg∠(G), where G is planar, drawings are 2D, bends are OK, but no crossings. seg×(G), where drawings are 2D, crossings are OK, but no bends and no overlaps.
seg2(K4) = 6 seg∠(K4) = 5 seg3(K3,3) = 7 seg×(K3,3) = 6
SLIDE 19
Relations Between Segment Number Variants
seg×(G) ≤ seg3(G)
SLIDE 20
Relations Between Segment Number Variants
seg×(G) ≤ seg3(G) seg3,×,∠(G) ≤ seg2(G) for any planar G.
SLIDE 21
Relations Between Segment Number Variants
seg×(G) ≤ seg3(G) seg3,×,∠(G) ≤ seg2(G) for any planar G. seg2(G)/ seg3,×,∠(G) = 2 + o(1) for a family of planar G.
SLIDE 22
Relations Between Segment Number Variants
seg×(G) ≤ seg3(G) seg3,×,∠(G) ≤ seg2(G) for any planar G. seg2(G)/ seg3,×,∠(G) = 2 + o(1) for a family of planar G.
SLIDE 23
Relations Between Segment Number Variants
seg×(G) ≤ seg3(G) seg3,×,∠(G) ≤ seg2(G) for any planar G. seg2(G)/ seg3,×,∠(G) = 2 + o(1) for a family of planar G.
SLIDE 24
Relations Between Segment Number Variants
seg×(G) ≤ seg3(G) seg3,×,∠(G) ≤ seg2(G) for any planar G. seg2(G)/ seg3,×,∠(G) = 2 + o(1) for a family of planar G.
SLIDE 25
Relations Between Segment Number Variants
seg×(G) ≤ seg3(G) seg3,×,∠(G) ≤ seg2(G) for any planar G. seg2(G)/ seg3,×,∠(G) = 2 + o(1) for a family of planar G. Open Problem. Find upper bounds for seg2(G)/ seg3,×,∠(G) for planar G.
SLIDE 26
Relations Between Segment Number Variants
SLIDE 27
Relations Between Segment Number Variants
Gk
SLIDE 28
Relations Between Segment Number Variants
×
Gk K2,3
SLIDE 29
Relations Between Segment Number Variants
= ×
Gk K2,3 G
SLIDE 30
Relations Between Segment Number Variants
= ×
Gk K2,3 G seg3(G) seg×(G) = 7k/2 5k/2+3 → 7 5.
SLIDE 31
Relations Between Segment Number Variants
= ×
Gk K2,3 G seg3(G) seg×(G) = 7k/2 5k/2+3 → 7 5. Open Problem. Can you do better?
SLIDE 32
Bounds on segment numbers of cubic graphs
G is a cubic graph with n ≥ 6 vertices. n/2 ≤ seg2,3,∠,×(G) ≤ 3n/2 and seg2,3,∠,×(⊔K4) = 3n/2.
SLIDE 33
Bounds on segment numbers of cubic graphs
γ seg2(G)∗ seg3(G) seg∠(G)∗ seg×(G) 1 5n/6..3n/2 5n/6∗..7n/5 5n/6..3n/2 5n/6∗..7n/5 2 3n/4..3n/2 5n/6.. 7n/5 3n/4..n + 1 3n/4∗..n + 2 3 n/2 + 3∗∗ 7n/10..7n/5 n/2 + 3 n/2.. n + 2 H 3n/4..3n/2 5n/6..n + 1 3n/4..n + 1 3n/4∗..n + 2 G is a cubic graph with n ≥ 6 vertices. n/2 ≤ seg2,3,∠,×(G) ≤ 3n/2 and seg2,3,∠,×(⊔K4) = 3n/2. * For planar G. ** by [Durocher et al. 2013; Igamberdiev et al. 2017]
SLIDE 34
Computational Complexity
Given a planar graph G, it is ∃R-hard to compute the slope number slope(G). [Hoffmann 2017]
SLIDE 35 Computational Complexity
Given a planar graph G and an integer k, it is ∃R-hard to decide whether ρ1
2(G) ≤ k and whether ρ1 3(G) ≤ k.
[Chaplick et al. 2017] Given a planar graph G, it is ∃R-hard to compute the slope number slope(G). [Hoffmann 2017]
SLIDE 36 Computational Complexity
Given a planar graph G and an integer k, it is ∃R-hard to decide whether ρ1
2(G) ≤ k and whether ρ1 3(G) ≤ k.
[Chaplick et al. 2017] Given a planar graph G, it is ∃R-hard to compute the slope number slope(G). [Hoffmann 2017] Given a planar graph G and an integer k, it is ∃R-complete to decide whether
SLIDE 37 Computational Complexity
Given a planar graph G and an integer k, it is ∃R-hard to decide whether ρ1
2(G) ≤ k and whether ρ1 3(G) ≤ k.
[Chaplick et al. 2017] Given a planar graph G, it is ∃R-hard to compute the slope number slope(G). [Hoffmann 2017] Given a planar graph G and an integer k, it is ∃R-complete to decide whether
- seg2(G) ≤ k,
- seg3(G) ≤ k,
- seg∠(G) ≤ k,
- seg×(G) ≤ k.
SLIDE 38
Arrangement Graphs
SLIDE 39
Arrangement Graphs
Arrangement graph G
SLIDE 40
Arrangement Graphs
Arrangement graph G Augmented arrangement graph G ′
SLIDE 41
Computational Complexity
The Arrangement Graph Recognition problem is to decide whether a given graph is the arrangement graph of some set of lines.
SLIDE 42
Computational Complexity
The Arrangement Graph Recognition problem is to decide whether a given graph is the arrangement graph of some set of lines. [Eppstein 2014] It is ∃R-complete.
SLIDE 43 Computational Complexity
The Arrangement Graph Recognition problem is to decide whether a given graph is the arrangement graph of some set of lines. Euclidean Pseudoline Stretchability is ∃R-hard. [Eppstein 2014]
⇑
[Matouˇ sek 2014, Schaefer 2009] It is ∃R-complete. A planar graph G is an arrangement graph on k lines ⇔ ρ1
2(G ′) ≤ k
⇔ seg2(G ′) ≤ k ⇔ seg∠(G ′) ≤ k ⇔ seg×(G ′) ≤ k. Open problem. Is any variant of segment number FPT? [Chaplick et al. 2017]
SLIDE 44
Lower Bounds for Cubic Graphs
SLIDE 45
Lower Bounds for Cubic Graphs
Flat vertex (f )
SLIDE 46
Lower Bounds for Cubic Graphs
Tripod vertex (t) Flat vertex (f )
SLIDE 47
Lower Bounds for Cubic Graphs
Tripod vertex (t) Bend (b) Flat vertex (f ) Lemma.
SLIDE 48
Lower Bounds for Cubic Graphs
For any straight-line drawing δ of a cubic graph with n vertices, seg(δ) = n/2 + t(δ) + b(δ). Tripod vertex (t) Bend (b) Flat vertex (f ) Lemma.
SLIDE 49
Connected Cubic Graphs
For any cubic connected graph G with n ≥ 6 vertices, seg3(G) ≤ 7n/5.
SLIDE 50
Connected Cubic Graphs
· · · For any cubic connected graph G with n ≥ 6 vertices, seg3(G) ≤ 7n/5.
SLIDE 51
Connected Cubic Graphs
· · · For any cubic connected graph G with n ≥ 6 vertices, seg3(G) ≤ 7n/5. seg2,3,∠,×(G) = 5k − 1 > 5n/6 n = 6k − 2
SLIDE 52
Biconnected Cubic Graphs
For any cubic biconnected planar graph G with n vertices, seg∠(G) ≤ n + 1. A corresponding drawing can be found in linear time.
SLIDE 53
Biconnected Cubic Graphs
For any cubic biconnected planar graph G with n vertices, seg∠(G) ≤ n + 1. A corresponding drawing can be found in linear time. [Liu et al. 1994]
SLIDE 54 Biconnected Cubic Graphs
For any cubic biconnected planar graph G with n vertices, seg∠(G) ≤ n + 1. A corresponding drawing can be found in linear time. Open Problem. What about 4-regular graphs? They have 2n
- edges. If we bend every edge once, we already need 2n
segments – and not all 4-regular graphs can be drawn with at most one bend per edge. [Liu et al. 1994]
SLIDE 55
Hamiltonian Cubic Graphs
For any cubic Hamiltonian graph G with n ≥ 6 vertices, seg3(G) ≤ n + 1.
SLIDE 56
Hamiltonian Cubic Graphs
For any cubic Hamiltonian graph G with n ≥ 6 vertices, seg3(G) ≤ n + 1. n = 4k seg2,∠,3,×(G) = 3n/4.
SLIDE 57
Hamiltonian Cubic Graphs
Each subgraph K ′ has an extreme point of its convex hull not connected to G − V (K ′). It is a tripod or a bend, so t(δ) + b(δ) ≥ k and, by Lemma, seg2,3,∠,×(G) ≥ 2k + t(δ) + b(δ) ≥ 3k . For any cubic Hamiltonian graph G with n ≥ 6 vertices, seg3(G) ≤ n + 1. n = 4k seg2,∠,3,×(G) = 3n/4.
SLIDE 58
Hamiltonian Cubic Graphs
Each subgraph K ′ has an extreme point of its convex hull not connected to G − V (K ′). It is a tripod or a bend, so t(δ) + b(δ) ≥ k and, by Lemma, seg2,3,∠,×(G) ≥ 2k + t(δ) + b(δ) ≥ 3k . For any cubic Hamiltonian graph G with n ≥ 6 vertices, seg3(G) ≤ n + 1. n = 4k seg2,∠,3,×(G) = 3n/4.
SLIDE 59
Hamiltonian Cubic Graphs
k ≥ 3, n = 6k, seg3(G) = 5n/6, seg×(G) = 2n/3
SLIDE 60
Hamiltonian Cubic Graphs
k ≥ 3, n = 6k, seg3(G) = 5n/6, seg×(G) = 2n/3
SLIDE 61
Open Problems: Improve Non-tight Bounds!
γ seg2(G)∗ seg3(G) seg∠(G)∗ seg×(G) 1 5n/6..3n/2 5n/6∗..7n/5 5n/6..3n/2 5n/6∗..7n/5 2 3n/4..3n/2 5n/6.. 7n/5 3n/4..n + 1 3n/4∗..n + 2 3 n/2 + 3∗∗ 7n/10..7n/5 n/2 + 3 n/2.. n + 2 H 3n/4..3n/2 5n/6..n + 1 3n/4..n + 1 3n/4∗..n + 2 G is a cubic graph with n ≥ 6 vertices. n/2 ≤ seg2,3,∠,×(G) ≤ 3n/2 and seg2,3,∠,×(⊔K4) = 3n/2. * For planar G. ** by [Durocher et al. 2013; Igamberdiev et al. 2017]
SLIDE 62
Open Problems: Improve Non-tight Bounds!
γ seg2(G)∗ seg3(G) seg∠(G)∗ seg×(G) 1 5n/6..3n/2 5n/6∗..7n/5 5n/6..3n/2 5n/6∗..7n/5 2 3n/4..3n/2 5n/6.. 7n/5 3n/4..n + 1 3n/4∗..n + 2 3 n/2 + 3∗∗ 7n/10..7n/5 n/2 + 3 n/2.. n + 2 H 3n/4..3n/2 5n/6..n + 1 3n/4..n + 1 3n/4∗..n + 2 G is a cubic graph with n ≥ 6 vertices. n/2 ≤ seg2,3,∠,×(G) ≤ 3n/2 and seg2,3,∠,×(⊔K4) = 3n/2. * For planar G. ** by [Durocher et al. 2013; Igamberdiev et al. 2017]
