IC-planar Graphs Recognizing and Drawing · Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg, Walter Didimo, William S. Evans,

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.


1-planar Graphs



1-planar Graphs



≤ 4n− 8 edges

1-planar Graphs



≤ 4n− 8 edges

straight-line: ≤ 4n− 9 edges

1-planar Graphs



≤ 4n− 8 edges


Recognition: NP-hard [Grigoriev & Bodlander ALG’07]

1-planar Graphs



≤ 4n− 8 edges



- for planar graphs + 1 edge [Korzhik & Mohar JGT’13]

1-planar Graphs



≤ 4n− 8 edges



- 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


RAC graphs:


... even for planar graphs [van Krefeld GD’11]


RAC Graphs


RAC graphs:



≤ 4n− 10 edges [Didimo et al. WADS’09]


RAC Graphs


RAC graphs:



≤ 4n− 10 edges [Didimo et al. WADS’09]


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


? RAC

1-planar RAC graphs

1-planar


RAC

outer-1-planar

outer-1-planar ⊂ RAC [Dehkordi & Eades IJCGA’12]

1-planar RAC graphs

1-planar


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


ndentcrossin

gs

IC-planar Graphs

≤ 13n/4− 6 edges


ndentcrossin

gs

Recognition

Reduction from 1-planarity testing.

Recognition


uv

Recognition


uv

Recognition


uv

Recognition


uv

u

Recognition


uv

u

Recognition


uv

u

Recognition

Testing IC-planarity is NP-hardTheorem.


uv

u

Recognition



uv

Reduction from planar-3SAT

Recognition

Recognition



uv


Recognition

Recognition


even if the rotation system is given.


uv


Recognition

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!



Task: Find a valid routing for each matching edge!Compute extended dual T ∗ of T .




T :

Compute extended dual T ∗ of T .




T :





T :





T :


T ∗ :




T :


u

v

T ∗ :

(u, v) ∈ EM




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v




T :


u

v

T ∗ :

(u, v) ∈ EM

u

v

Routing in T = path of length 3 in T ∗



u

v

luv ruv



u

v

luv ruv

Interior I(u, v)



u

v

luv ruv

Interior I(u, v)The boundaries of twointeriors may not intersect.



u

v

luv ruv


X



u

v

luv ruv


X

X



u

v

luv ruv


X

X

X



u

v

luv ruv


X

X

X

×



u

v


u

v


u

v


a

b

u

v


a

b

u

v


a

b

u

v


a

b

c d

u

v


a

b

c d

u

v


a

b

c d

u

v


a

b

c d

u

v

Hierarchical structure: Tree H = (VH , EH)

H:


a

b

c d

u

vIcdIab

Iuv

Hierarchical structure: Tree H = (VH , EH)VH = {Iuv | (u, v) ∈M}

H:


a

b

c d

u

vIcdIab

Iuv

G

Hierarchical structure: Tree H = (VH , EH)VH = {Iuv | (u, v) ∈M} ∪ {G}

H:


a

b

c d

u

vIcdIab

Iuv

G


(Iuv, Iab) ∈ EH ⇔ Iuv ⊂ IabVH = {Iuv | (u, v) ∈M} ∪ {G}

H:


a

b

c d

u

vIcdIab

Iuv

G


(Iuv, Iab) ∈ EH ⇔ Iuv ⊂ Iaboutdeg(Iuv) = 0⇒ (Iuv, G) ∈ EH

VH = {Iuv | (u, v) ∈M} ∪ {G}

H:


a

b

c d

u

vIcdIab

Iuv

G


a

b

c dIcdIab

Iuv

G


a

b

c dIcdIab

Iuv

G


a

b

c dIcdIab

Iuv

G


a

b

c dIcdIab

Iuv

G


a

b

c dIcdIab

Iuv

G


a

b

c dIcdIab

Iuv

G


Always pick “middle” routing

a

b

c dIcdIab

Iuv

G



a

b

Solve rest with 2SAT

c dIcdIab

Iuv

G



a

b


c d

u

vIcdIab

Iuv

G



a

b


c d

u

vIcdIab

Iuv

G

Recursively check which routings are valid



a

b


c d

u

vIcdIab

Iuv

G




a

b


c d

u

vIcdIab

Iuv

G




a

b


c d

u

vIcdIab

Iuv

G




a

b


c d

u

vIcdIab

Iuv

G


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.



Theorem.

Using a special 1-planar drawing...

[Alam et al. GD’13]



Theorem.


RAC?




Theorem.


RAC?




Theorem.

Straight-line RAC drawings of IC-planar graphs may requireexponential area.

Theorem.


RAC?




Theorem.

Straight-line RAC drawings of IC-planar graphs may requireexponential area.

Theorem.


RAC?


Straight-Line RAC Drawings

Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]


Adjust Shift-Algorithm for planar graphs

Augment to 3-connected planar graph

[de Fraysseix, Pach & Pollack Comb’90]



Augment to 3-connected planar graphInsert vertices in canonical order





Contour only has slopes ±1Insert vertices in canonical order









































Augment to planar-maximal IC-planar graph





d

cb

a


Each crossing → Kite K = (a, b, c, d)


Remove one edge per crossing






d

cb

a


Adjust step in which d is placed







d

cb

a









d

cb

a

Highest number incanonical order









d

cb

a

c

a

Al(b) b



d

cb

a

c

a

Al(b) b



d

cb

a

c

a

Al(b) b



d

cb

a

c

a

Al(b) b



d

cb

a

c

a

Al(b) b



d

cb

a

c

a



d

cb

a

Al(b) b c

a



d

cb

a

Al(b) b c

a



d

cb

a

Al(b) b c

a

d



d

cb

a

Al(b) b c

a

d



d

cb

a

Al(b) c

a

d

b



d

cb

a

Al(b) b

a

c



d

cb

a

Al(b) bu

d

a

c



d

cb

a

Al(b) bu

d

a

c



d

cb

a

Al(b) bu

d

a

c



d

cb

a

Al(b) bu

d

a

c



d

cb

a

Al(b) bu

d

a

c



d

cb

a

bu

d

a

c



d

cb

a

bu

d

a

c



d

cb

a

bu

c

a



d

cb

a

u

c

a

b



d

cb

a

u

c

a

b



d

cb

a

u

c

a

Al(b) b



d

cb

a

u

c

a

Al(b) b



d

cb

a

u

c

a

Al(b) b



d

cb

a

cAl(b) ba

u



d

cb

a

cAl(b) b

d

a

u



d

cb

a

cAl(b) b

d

a

u



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?

IC-planar Graphs Recognizing and Drawing · Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg, Walter Didimo, William S. Evans,

Documents