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Recognizing and Drawing
IC-planar Graphs
Philipp KindermannUniversitat Wurzburg /
FernUniversitat in Hagen
Joint work withFranz J. Brandenburg, Walter Didimo, William S. Evans,
Giuseppe Liotta & Fabrizio Montecchiani
1-planar Graphs
Planar graphs: Can be drawn without crossings.
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges• Recognition: NP-hard [Grigoriev & Bodlander ALG’07]
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges• Recognition: NP-hard [Grigoriev & Bodlander ALG’07]
- for planar graphs + 1 edge [Korzhik & Mohar JGT’13]
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges• Recognition: NP-hard [Grigoriev & Bodlander ALG’07]
- for planar graphs + 1 edge [Korzhik & Mohar JGT’13]
- with given rotation system [Auer et al. JGAA’15]
RAC Graphs
RAC graphs: Can be drawn straight-linewith only right-angle crossings.
RAC Graphs
RAC graphs: Can be drawn straight-linewith only right-angle crossings.
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
Can be drawn straight-linewith only right-angle crossings.
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
... even for planar graphs [van Krefeld GD’11]
Can be drawn straight-linewith only right-angle crossings.
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
... even for planar graphs [van Krefeld GD’11]
• ≤ 4n− 10 edges [Didimo et al. WADS’09]
Can be drawn straight-linewith only right-angle crossings.
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
... even for planar graphs [van Krefeld GD’11]
• ≤ 4n− 10 edges [Didimo et al. WADS’09]
Can be drawn straight-linewith only right-angle crossings.
• Recognition: NP-hard [Argyriou et al. JGAA’12]
1-planar RAC graphs
1-planar
1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
RAC
1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
? RAC
1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
RAC
outer-1-planar
• outer-1-planar ⊂ RAC [Dehkordi & Eades IJCGA’12]
1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
RAC
outer-1-planar
• outer-1-planar ⊂ RAC [Dehkordi & Eades IJCGA’12]
perfect RAC
• perfect RAC ⊂ 1-planar [Eades & Liotta DMA’13]
IC-planar Graphs
IC-planar graphs: Each edge is crossed at most once
independent
crossings
IC-planar Graphs
IC-planar graphs: Each edge is crossed at most onceand each vertex is incident toat most one crossing edge.indepe
ndentcrossin
gs
IC-planar Graphs
IC-planar graphs: Each edge is crossed at most onceand each vertex is incident toat most one crossing edge.indepe
ndentcrossin
gs
IC-planar Graphs
• ≤ 13n/4− 6 edges
IC-planar graphs: Each edge is crossed at most onceand each vertex is incident toat most one crossing edge.indepe
ndentcrossin
gs
Recognition
Reduction from 1-planarity testing.
Recognition
Reduction from 1-planarity testing.
uv
Recognition
Reduction from 1-planarity testing.
uv
Recognition
Reduction from 1-planarity testing.
uv
Recognition
Reduction from 1-planarity testing.
uv
u
Recognition
Reduction from 1-planarity testing.
uv
u
Recognition
Reduction from 1-planarity testing.
uv
u
Recognition
Testing IC-planarity is NP-hardTheorem.
Reduction from 1-planarity testing.
uv
u
Recognition
Testing IC-planarity is NP-hardTheorem.
Reduction from 1-planarity testing.
uv
Reduction from planar-3SAT
Recognition
Recognition
Testing IC-planarity is NP-hardTheorem.
Reduction from 1-planarity testing.
uv
Reduction from planar-3SAT
Recognition
Recognition
Testing IC-planarity is NP-hardTheorem.
even if the rotation system is given.
Reduction from 1-planarity testing.
uv
Reduction from planar-3SAT
Recognition
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!Compute extended dual T ∗ of T .
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
T ∗ :
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Routing in T = path of length 3 in T ∗
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
X
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
X
X
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
X
X
×
Triangulation + Matching
Triangulation + Matching
u
v
Triangulation + Matching
u
v
Triangulation + Matching
u
v
Triangulation + Matching
a
b
u
v
Triangulation + Matching
a
b
u
v
Triangulation + Matching
a
b
u
v
Triangulation + Matching
a
b
c d
u
v
Triangulation + Matching
a
b
c d
u
v
Triangulation + Matching
a
b
c d
u
v
Triangulation + Matching
a
b
c d
u
v
Hierarchical structure: Tree H = (VH , EH)
H:
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
Hierarchical structure: Tree H = (VH , EH)VH = {Iuv | (u, v) ∈M}
H:
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
Hierarchical structure: Tree H = (VH , EH)VH = {Iuv | (u, v) ∈M} ∪ {G}
H:
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
Hierarchical structure: Tree H = (VH , EH)
(Iuv, Iab) ∈ EH ⇔ Iuv ⊂ IabVH = {Iuv | (u, v) ∈M} ∪ {G}
H:
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
Hierarchical structure: Tree H = (VH , EH)
(Iuv, Iab) ∈ EH ⇔ Iuv ⊂ Iaboutdeg(Iuv) = 0⇒ (Iuv, G) ∈ EH
VH = {Iuv | (u, v) ∈M} ∪ {G}
H:
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
Triangulation + Matching
• Always pick “middle” routing
a
b
c dIcdIab
Iuv
G
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c dIcdIab
Iuv
G
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
Theorem.IC-planarity can be tested efficiently if the input graph is atriangulated planar graph and a matching
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Using a special 1-planar drawing...
[Alam et al. GD’13]
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Straight-line RAC drawings of IC-planar graphs may requireexponential area.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Straight-line RAC drawings of IC-planar graphs may requireexponential area.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
Straight-Line RAC Drawings
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
Straight-Line RAC Drawings
• Adjust step in which d is placed
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
Straight-Line RAC Drawings
• Adjust step in which d is placed
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
Highest number incanonical order
Straight-Line RAC Drawings
• Adjust step in which d is placed
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
d
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
d
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) c
a
d
b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
bu
d
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
bu
d
a
c
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
bu
c
a
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
Al(b) b
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) ba
u
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) b
d
a
u
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) b
d
a
u
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) b
d
a
uIC-planar graphs can be drawn straight-line RAC inexponential area in O(n3) time.
Theorem.
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
?
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
?
Draw in polynomial area with good crossing resolution?
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
?
Draw in polynomial area with good crossing resolution?
What about maximal IC-planar graphs?
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