[PPT] - Planar Octilinear Drawings with One Bend Per Edge M. A. Bekos 1 , M. PowerPoint Presentation

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Planar Octilinear Drawings with One Bend Per Edge

M. A. Bekos1, M. Gronemann2, M. Kaufmann1, R. Krug1

1Wilhelm Schickard Institut f¨

ur Informatik, Universit¨ at T¨ ubingen, Germany

2Institut fur Informatik, Universit¨

at zu K¨

ln, Germany

26.09.2014

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Motivation

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Previous- and Related Work

M. N¨
llenburg: Automated drawings of metro maps [2005]

NP-hard if 0 bends is possible

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Previous- and Related Work

M. N¨
llenburg: Automated drawings of metro maps [2005]

NP-hard if 0 bends is possible

B. Keszegh et al.: Drawing planar graphs of bounded degree with

few slopes [2013]

maxdeg. 8 with 2 bends

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Previous- and Related Work

M. N¨
llenburg: Automated drawings of metro maps [2005]

NP-hard if 0 bends is possible

B. Keszegh et al.: Drawing planar graphs of bounded degree with

few slopes [2013]

maxdeg. 8 with 2 bends
E. Di Giacomo et al.: The planar slope number of subcubic

graphs [2014]

maxdeg. 3 with 0 bends

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph k-connected graph

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph k-connected graph Canonical ordering (for triconnected graphs)

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph k-connected graph Canonical ordering (for triconnected graphs)

Partitioning of G into m paths with P0 = {v1,v2} and Pm = {vn} such that:

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph k-connected graph Canonical ordering (for triconnected graphs)

Partitioning of G into m paths with P0 = {v1,v2} and Pm = {vn} such that: Gk is biconnected

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph k-connected graph Canonical ordering (for triconnected graphs)

Partitioning of G into m paths with P0 = {v1,v2} and Pm = {vn} such that: Gk is biconnected All neighbors of Pk+1 in Gk are on the outer face of Gk

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph k-connected graph Canonical ordering (for triconnected graphs)

Partitioning of G into m paths with P0 = {v1,v2} and Pm = {vn} such that: Gk is biconnected All neighbors of Pk+1 in Gk are on the outer face of Gk All vertices of Pk have at least one neighbor in a Pj with j > k

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Preliminaries

k-planar graph k-connected graph Canonical ordering (for triconnected graphs)

Partitioning of G into m paths with P0 = {v1,v2} and Pm = {vn} such that: Gk is biconnected All neighbors of Pk+1 in Gk are on the outer face of Gk All vertices of Pk have at least one neighbor in a Pj with j > k

|Pk| = 1 is called singleton, |Pk| > 1 is called chain

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v1 v2

Start of the construction

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v1 v2 v3 v|P1|+2

First Partition

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v1 v2 vi vj v′

i

v′

j Placing a chain may require stretching

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v1 v2 vi vj v′

i

v′

j

Placing a chain

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

vi v1 v2 v

Placing a singleton

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

vn v1 v2 v3

Placing of vn step 1

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v2 vn v3 v1 Final layout

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Results for 4-planar Graphs

Theorem There exists an infinite class of 4-planar graphs which do not admit bendless octilinear drawings and if they are drawn with at most one bend per edge, then a linear number of bends is required

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Results for 4-planar Graphs

Theorem There exists an infinite class of 4-planar graphs which do not admit bendless octilinear drawings and if they are drawn with at most one bend per edge, then a linear number of bends is required Theorem Given a triconnected 4-planar graph G, we can compute in O(n) time an octilinear drawing of G with at most one bend per edge on an O(n2)× O(n) integer grid.

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Non-triconnected Graphs

Extend to biconnected by using SPQR-trees and the triconnected algorithm for the R-nodes

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Non-triconnected Graphs

Extend to biconnected by using SPQR-trees and the triconnected algorithm for the R-nodes Extend to connected using the BC-tree and the biconnected algorithm

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v1 v2

Start of the construction

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v1 v2 v3 v|P1|+2 First Partition

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

v1 v2 vi vj v′

i

v′

j

Placing a chain

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

vi v1 v2 v v′

Placing a singleton

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

The Triconnected Case

vn v1 v2 v3

Final layout

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Bad news

v2 v1 Gn Gn+1

Super-polynomial area requirement h(Gn) > w(Gn) w(Gn+1) ≥ 2w(Gn) w(Gn+1) ≥ h(Gn) h(Gn+1) ≥ 2h(Gn)

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Properties of the 5-planar Algorithm

Theorem Given a triconnected 5-planar graph G, we can compute in O(n2) time an octilinear drawing of G with at most one bend per edge.

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Non-triconnected Graphs

Extend to biconnected by using SPQR-trees and the triconnected algorithm for the R-nodes

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Non-triconnected Graphs

Extend to biconnected by using SPQR-trees and the triconnected algorithm for the R-nodes Extend to connected using the BC-tree and the biconnected algorithm

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

One Bend Per Edge Is Not Always Enough

v1 v2 v3

Outer Face that does not admit a one-bend drawing

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

One Bend Per Edge Is Not Always Enough

6-planar triangulation in which each is adjacent to only degree 6 (grey) vertices and at most one degree 5 (black) vertex

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

One Bend Per Edge Is Not Always Enough

f1 G1 f2 G2 Gaug

1

⊕ Gaug

2

f ′

1

f ′

2

Gaug

1

Gaug

2

f ′

1

Γ(Gaug

2

)

Construction of an infinite family of graphs

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

One Bend Per Edge Is Not Always Enough

f1 G1 f2 G2 Gaug

1

⊕ Gaug

2

f ′

1

f ′

2

Gaug

1

Gaug

2

f ′

1

Γ(Gaug

2

)

Construction of an infinite family of graphs

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

One Bend Per Edge Is Not Always Enough

f1 G1 f2 G2 Gaug

1

⊕ Gaug

2

f ′

1

f ′

2

Gaug

1

Gaug

2

f ′

1

Γ(Gaug

2

)

Construction of an infinite family of graphs

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

One Bend Per Edge Is Not Always Enough

f1 G1 f2 G2 Gaug

1

⊕ Gaug

2

f ′

1

f ′

2

Gaug

1

Gaug

2

f ′

1

Γ(Gaug

2

)

Construction of an infinite family of graphs

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Conclusion

4-planar graphs are octilinear drawable with at most one bend per edge in cubic area in linear time

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Conclusion

4-planar graphs are octilinear drawable with at most one bend per edge in cubic area in linear time 5-planar graphs are octilinear drawable with at most one bend per edge in super-polynomial area in quadratic time

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Conclusion

4-planar graphs are octilinear drawable with at most one bend per edge in cubic area in linear time 5-planar graphs are octilinear drawable with at most one bend per edge in super-polynomial area in quadratic time There exist 6-planar graphs that do not admit planar octilinear drawings with at most one bend per edge

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Open Problems

Is it possible to have 4-planar octilinear drawings in less than O(n3) area?

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Open Problems

Is it possible to have 4-planar octilinear drawings in less than O(n3) area? What is the area requirement of 5-planar (triconnected) graphs?

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Open Problems

Is it possible to have 4-planar octilinear drawings in less than O(n3) area? What is the area requirement of 5-planar (triconnected) graphs? Do triangle-free 6-planar graph admit one-bend octilinear drawings?

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Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion

Open Problems

Is it possible to have 4-planar octilinear drawings in less than O(n3) area? What is the area requirement of 5-planar (triconnected) graphs? Do triangle-free 6-planar graph admit one-bend octilinear drawings? What is the complexity to determine whether a 6-planar graph admits a one-bend octilinear drawing?