CCCG 2014

CCCG 2014 August 11, 2014

Drawing Plane Triangulations with Few Segments

Department of Computer ScienceUniversity of Manitoba

Stephane Durocher Debajyoti Mondal

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k-Segment Drawings

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CCCG 2014

August 11, 2014

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k-Segment Drawings

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CCCG 2014

August 11, 2014

a b

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iA segment is maximal path P such that the vertices on P

are collinear in the drawing.

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k-Segment Drawings

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(a) (b)

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CCCG 2014

August 11, 2014

A 8-segment drawingA 10-segment drawing

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k-Segment Drawings

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CCCG 2014

August 11, 2014

A 8-segment drawingA 10-segment drawing

Minimization is NP-complete [Durocher, Mondal, Nishat,

and Whitesides, CCCG 2011]

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Previous Results

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August 11, 2014

Graph Class Lower Bounds Upper Bounds References

Trees | {v: deg(v) is odd} | / 2 | {v: deg(v) is odd} | / 2

Dujmović, Eppstein, Suderman and Wood,

CGTA 2007Maximal Outerplanar n n

Plane 2-Trees and 3-Trees 2n 2n

3-Connected Cubic Plane n/2+3 n/2+4Mondal, Nishat,

Biswas, and Rahman, JOCO 2010

3-Connected Plane 2n 5n/2 (= 2.50n)Dujmović, Eppstein, Suderman and Wood,

CGTA 2007

Triangulations 2n 7n/3 (= 2.33n) This Presentation

4-Conneted Triangulations X 9n/4 (= 2.25n) This Presentation

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Better Upper Bound for Triangulations

CCCG 2014

August 11, 2014

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Every rooted tree T has an upward drawing with leaf (T) segments.

Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

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CCCG 2014

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Whenever we create a new segment, we ensure that this new segment starts at a non-leaf vertex T and ends at a leaf of T The drawing has leaf (T) segments.

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CCCG 2014

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Whenever we create a new segment, we ensure that this new segment starts at a non-leaf vertex T and ends at a leaf of T The drawing has leaf (T) segments.

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Divergence: downward extension of the segments does not create edge

crossings.


Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees

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A Canonical Ordering of G[De Fraysseix, Pach, and Pollack 1988]


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A Canonical Ordering of G[De Fraysseix et al. 1988]

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A Schnyder realizer of G [Schnyder 1990]


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A Canonical Ordering of G[De Fraysseix et al. 1988]

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A Schnyder realizer of G [Schnyder 1990]

leaf (Tl) =3v7

leaf (Tr) =3

Tm

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Draw G with at most leaf (Tl) + leaf (Tr) + n segments

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Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees

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leaf (Tl) + leaf (Tr) + 3 segments



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leaf (Tl) + leaf (Tr) + 3 segments+ at most (n-3) segments

28CCCG 2014

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Why Does this Work?

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Question 1. What makes it possible to maintain nice drawings of the subtrees?

- We can always create a new segment satisfying the ‘divergence’ property.

A triangulation G and A drawing of G with at most leaf (Tl) + leaf (Tr) + n segments

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29CCCG 2014

August 11, 2014

Why Does this Work?

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Question 1. What makes it possible to maintain nice drawings of the subtrees?

- We can always create a new segment satisfying the ‘divergence’ property.

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Why Does this Work?

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vQuestion 2. Why the drawing of the edges in Tm does not create any edge crossing?

- The slopes of the l-edges incident to the outerface are smaller than the slope of edge (v, p).- The slopes of the r-edges incident to the outerface are larger than the slope of edge (v, q).

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Final Upper Bounds

CCCG 2014

August 11, 2014

Graph Class Lower Bounds Upper Bounds References

Trees| {v: deg(v) is odd} | / 2

| {v: deg(v) is odd} | / 2 Dujmović, Eppstein, Suderman and Wood,

CGTA 2007Maximal Outerplanar n n

Plane 2-Trees and 3-Trees

2n 2n Samee, Alam, Adnan and Rahman, GD 2008

3-Connected Cubic Plane Graphs

n/2 n/2Mondal, Nishat,

Biswas, and Rahman, JOCO 2010

3-Connected Plane Graphs

2n 5n/2 (= 2.50n)Dujmović, Eppstein, Suderman and Wood,

CGTA 2007

Triangulations 2n leaf (Tl) + leaf (Tr) + n <= 2.33n This Presentation

4-Conneted Triangulations X leaf (Tl) + leaf (Tr) + n <= 2.25n This Presentation

Combine the upper bounds on the number of leaves

[Bonichon, Saëc and Mosbah, ICALP 2002]

[Zhang and He, DCG 2005]

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Future Research

CCCG 2014

August 11, 2014

Tight Bounds: What is the smallest constant c such that every n vertex planar graph admits a (cn)-segment drawing? Can we improve the bound in the variable embedding setting?

Generalization: Does the upper bound of 7n/3 segments hold also for 3-connected planar graphs?

Optimization: Is there a polynomial-time algorithm for computing minimum-segment drawings of triangulations, or simpler classes of graphs such as plane 3-trees or outerplanar graphs?

Thank You..

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Tight Bounds?

Generalization?

Optimization?

CCCG 2014

Documents

Transcript of CCCG 2014