SLIDE 1 Drawing Subcubic 1-Planar Graphs with Few Bends, Few Slopes, and Large Angles
Philipp Kindermann Cheriton School of Computer Science University of Waterloo
Joint work with Fabrizio Montecchiani, Lena Schlipf, and Andr´ e Schulz
SLIDE 2 Slope Number
“How many slopes do I need to draw a graph straight-line?”
SLIDE 3 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes
SLIDE 4 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?”
SLIDE 5 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes
SLIDE 6 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?”
SLIDE 7 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes
SLIDE 8 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes
SLIDE 9 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes maximum degree ∆
SLIDE 10 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes maximum degree ∆ unbounded w.r.t. ∆ [Bar´ at, Matousek, Wood 2006] [Pach, P´ alv¨
SLIDE 11 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes maximum degree ∆ unbounded w.r.t. ∆ [Bar´ at, Matousek, Wood 2006] [Pach, P´ alv¨
bounded by 2O(∆) [Keszegh, Pach, P´ alv¨
SLIDE 12 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes maximum degree ∆ unbounded w.r.t. ∆ [Bar´ at, Matousek, Wood 2006] [Pach, P´ alv¨
2 bends: ≤ ⌈∆/2⌉ [Keszegh, Pach, P´ alv¨
bounded by 2O(∆) [Keszegh, Pach, P´ alv¨
SLIDE 13 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes maximum degree ∆ 1 bend: ≤ ∆ − 1 [Angelini, Bekos, Liotta, Montecchiani 2017] unbounded w.r.t. ∆ [Bar´ at, Matousek, Wood 2006] [Pach, P´ alv¨
2 bends: ≤ ⌈∆/2⌉ [Keszegh, Pach, P´ alv¨
bounded by 2O(∆) [Keszegh, Pach, P´ alv¨
SLIDE 14 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes maximum degree ∆ 1 bend: ≤ ∆ − 1 [Angelini, Bekos, Liotta, Montecchiani 2017] unbounded w.r.t. ∆ [Bar´ at, Matousek, Wood 2006] [Pach, P´ alv¨
2 bends: ≤ ⌈∆/2⌉ [Keszegh, Pach, P´ alv¨
subcubic: ≤ 4 [Di Giacomo, Liotta, Montec. 2018] bounded by 2O(∆) [Keszegh, Pach, P´ alv¨
SLIDE 15 Slope Number
“How many slopes do I need to draw a graph straight-line?” 4 slopes “How many slopes do I need to draw a graph planar straight-line?” 6 slopes “How many slopes do I need to draw a graph planar with k bends?” 1 bend, 3 slopes 2 bend, 2 slopes maximum degree ∆ 1 bend: ≤ ∆ − 1 [Angelini, Bekos, Liotta, Montecchiani 2017] unbounded w.r.t. ∆ [Bar´ at, Matousek, Wood 2006] [Pach, P´ alv¨
2 bends: ≤ ⌈∆/2⌉ [Keszegh, Pach, P´ alv¨
subcubic: ≤ 4 [Di Giacomo, Liotta, Montec. 2018] 1−? 1−? bounded by 2O(∆) [Keszegh, Pach, P´ alv¨
SLIDE 16
0 Bends, Lower Bounds
SLIDE 17
0 Bends, Lower Bounds
SLIDE 18
0 Bends, Lower Bounds
SLIDE 19
0 Bends, Lower Bounds
SLIDE 20
0 Bends, Lower Bounds
SLIDE 21
0 Bends, Lower Bounds
SLIDE 22
0 Bends, Lower Bounds
SLIDE 23
0 Bends, Lower Bounds
SLIDE 24
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 25
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 26
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 27
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 28
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 29
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 30
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 31
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 32
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 33
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 34
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes
SLIDE 35
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes 3-regular 3-connected 1-plane ⇒≥ 18 slopes
SLIDE 36
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes 3-regular 3-connected 1-plane ⇒≥ 18 slopes
SLIDE 37
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes 3-regular 3-connected 1-plane ⇒≥ 18 slopes
SLIDE 38
0 Bends, Lower Bounds
2-regular 2-connected 1-plane ⇒ Ω(n) slopes 3-regular 3-connected 1-plane ⇒≥ 18 slopes maxdeg-∆ 3-connected 1-plane ⇒≥ 9(∆ − 1) slopes
SLIDE 39
1 Bend, Lower Bound
SLIDE 40
1 Bend, Lower Bound
SLIDE 41
1 Bend, Lower Bound
SLIDE 42
1 Bend, Lower Bound
SLIDE 43
1 Bend, Lower Bound
cubic 3-connected 1-plane 1-bend ⇒ 3 slopes
SLIDE 44
Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
SLIDE 45
Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
SLIDE 46
Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
SLIDE 47 Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
A B
SLIDE 48 Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
A B A B
SLIDE 49 Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
A B A B
SLIDE 50 Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
A B A B A
SLIDE 51 Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
A B A B A A
SLIDE 52 Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
A B A B A A A
SLIDE 53 Reembedding 1-Planar Graphs
G 1-plane, G∗ planarization
⇒ re-embed G 1-plane s.t. no cutvertex of G∗ is dummy
and if G 3-connected, then G∗ 3-connected.
A B A B A A A A
SLIDE 54
Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
SLIDE 55
Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
SLIDE 56 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 57 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 58 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 59 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 60 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 61 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 62 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 63 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 64 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
SLIDE 65 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2
SLIDE 66 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
SLIDE 67 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
SLIDE 68 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex
SLIDE 69 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1
SLIDE 70 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain
SLIDE 71 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
SLIDE 72 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 73 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 74 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 75 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 76 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 77 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 78 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 79 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 80 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 81 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 82 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1 v2 v1
V2:
real vertex
V2
dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 83 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 84 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 85 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 86 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 87 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 88 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 89 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 90 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 91 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 92 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 93 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 94 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 95 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 96 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 97 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 98 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 99 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 100 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 101 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 102 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 103 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 104 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 105 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v2 v1
V2:
real vertex dummy vertex v2 v1 chain v2 v1
Vi:
SLIDE 106 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2
SLIDE 107 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 108 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 109 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 110 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 111 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 112 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 113 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 114 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
SLIDE 115 Cubic 3-Connected ⇒ 1 Bend, 4 Slopes
v1 v2 v1 v2
cubic 3-connected 1-plane 1-bend ⇒≤ 4 slopes
SLIDE 116 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn
SLIDE 117 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 118 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 119 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 120 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 121 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 122 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 123 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 124 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 125 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 126 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 127 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2
SLIDE 128 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 129 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 130 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 131 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 132 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 133 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 134 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 135 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 136 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn v1 v2 vn
SLIDE 137 Subcubic ⇒ 2 Bends, 2 Slopes
v1 v2 vn
subcubic 1-plane 2-bend ⇒≤ 2 slopes
v1 v2 vn
SLIDE 138
Conclusion
2-regular 2-connected 1-plane s-l ⇒ Ω(n) slopes
SLIDE 139
Conclusion
3-connected 1-plane s-l ⇒≥ 9(∆ − 1) slopes 2-regular 2-connected 1-plane s-l ⇒ Ω(n) slopes
SLIDE 140
Conclusion
cubic 3-connected 1-plane 1-bend ⇒≥ 3 slopes 3-connected 1-plane s-l ⇒≥ 9(∆ − 1) slopes 2-regular 2-connected 1-plane s-l ⇒ Ω(n) slopes
SLIDE 141
Conclusion
cubic 3-connected 1-plane 1-bend ⇒≥ 3 slopes cubic 3-connected 1-plane 1-bend ⇒≤ 4 slopes 3-connected 1-plane s-l ⇒≥ 9(∆ − 1) slopes 2-regular 2-connected 1-plane s-l ⇒ Ω(n) slopes
SLIDE 142
Conclusion
cubic 3-connected 1-plane 1-bend ⇒≥ 3 slopes cubic 3-connected 1-plane 1-bend ⇒≤ 4 slopes subcubic 1-plane 2-bend ⇒≤ 2 slopes 3-connected 1-plane s-l ⇒≥ 9(∆ − 1) slopes 2-regular 2-connected 1-plane s-l ⇒ Ω(n) slopes
SLIDE 143 Conclusion
cubic 3-connected 1-plane 1-bend ⇒≥ 3 slopes cubic 3-connected 1-plane 1-bend ⇒≤ 4 slopes subcubic 1-plane 2-bend ⇒≤ 2 slopes 3-connected 1-plane s-l ⇒≥ 9(∆ − 1) slopes
angular and crossing resolution: π/4 and π/2
2-regular 2-connected 1-plane s-l ⇒ Ω(n) slopes
SLIDE 144 Conclusion
cubic 3-connected 1-plane 1-bend ⇒≥ 3 slopes cubic 3-connected 1-plane 1-bend ⇒≤ 4 slopes subcubic 1-plane 2-bend ⇒≤ 2 slopes
?
3-connected 1-plane s-l ⇒≥ 9(∆ − 1) slopes
angular and crossing resolution: π/4 and π/2
2-regular 2-connected 1-plane s-l ⇒ Ω(n) slopes
