Balanced Circle Packings for Planar Graphs Md. Jawaherul Alam David - - PowerPoint PPT Presentation
University of California, Irvine University of Arizona Graph Drawing Wrzburg September 24, 2014 Balanced Circle Packings for Planar Graphs Md. Jawaherul Alam David Epqsuein Stepien G. Kobovrov Michael T . Goodsich Sergfy Pupyrev .
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■ Contact representation with circles
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■ Contact representation with circles ■ Vertices are interior-disjoint circles ■ Edges are contacts between circles
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■ Contact representation with circles ■ Vertices are interior-disjoint circles ■ Edges are contacts between circles
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7 6 5 3 2 8 1 9 4
3 2 9 1 4 5 8
■ Contact representation with circles ■ Vertices are interior-disjoint circles ■ Edges are contacts between circles
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■ Polynomial ratio between maximum and minimum diameter
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■ Any plane graph has a circle-packing [Koebe, 1936]. ■ Any 3-connected plane graph has a primal-dual circle packing
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■ Any plane graph has a circle-packing [Koebe, 1936]. ■ Any 3-connected plane graph has a primal-dual circle packing
■ It is NP-complete to test whether a graph admits contact
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■ Any plane graph has a circle-packing [Koebe, 1936]. ■ Any 3-connected plane graph has a primal-dual circle packing
■ It is NP-complete to test whether a graph admits contact
■ In a realization with integer radii, radius of 22Θ(n) is sometimes
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k d b c e f g l p h i j m n
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k d b c e f g l p h i j m n
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k d b c e f g l p h i j m n
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k d b c e f g l p h i j m n
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k d b c e f g l p h i j m n
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k d b c e f g l p h i j m n
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k d b c e f g l p h i j m n
b c d e f g h i j k l m n o p a ■ Compute balanced square-contact representation
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k d b c e f g l p h i j m n
b c d e f g h i j k l m n o p a ■ Compute balanced square-contact representation
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k d b c e f g l p h i j m n
b c d e f g h i j k l m n o p a a c d e f g l p h i m j k n o b ■ Compute balanced square-contact representation ■ Draw Inscribing circles inside the squares
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k d b c e f g l p h i j m n
b c d e f g h i j k l m n o p a a c d e f g l p h i m j k n o b ■ Compute balanced square-contact representation ■ Draw Inscribing circles inside the squares ■ Translate downwards
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k d b c e f g l p h i j m n
b c d e f g h i j k l m n o p a a m k n o j d h i b g f e l p c ■ Compute balanced square-contact representation ■ Draw Inscribing circles inside the squares ■ Translate downwards
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k d b c e f g l p h i j m n
k d b c e f g l p h i j m n
■ Add a path between the children of every vertex
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k d b c e f g l p h i j m n
k d b c e f g l p h i j m n
■ Add a path between the children of every vertex
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■ Follow the algorithm for balanced packing of the tree ■ Modify the circles for the children of each vertex
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■ Each cactus graph is a subgraph of an augmented fan-tree
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■ Each cactus graph is a subgraph of an augmented fan-tree
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■ Each cactus graph is a subgraph of an augmented fan-tree
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■ Draw Circles for spine vertices
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■ Draw Circles for spine vertices ■ Rotate to create space for other vertices
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■ G: maximal planar graph ■ ∆: maximum vertex-degree in G
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■ G: maximal planar graph ■ ∆: maximum vertex-degree in G
R ≥ α∆−2, α ≈ 0.15.
R r
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■ G: maximal planar graph ■ ∆: maximum vertex-degree in G
R ≥ α∆−2, α ≈ 0.15.
R r
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k log n diameter.
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■ Idea: Triangulate with O(∆) degree and k + log n diameter
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■ Idea: Triangulate with O(∆) degree and k + log n diameter
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■ Balanced circle packing for graphs with
■ Balanced circle packing for trees, cactus graphs and outerpaths ■ Balanced circle packing for graphs with bounded tree-depth
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■ Balanced circle packing for outerplanar graphs
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■ Balanced circle packing for outerplanar graphs
■ Balanced intersection representation