Balanced Circle Packings for Planar Graphs Md. Jawaherul Alam David - PowerPoint PPT Presentation

University of California, Irvine University of Arizona Graph Drawing Würzburg – September 24, 2014 Balanced Circle Packings for Planar Graphs Md. Jawaherul Alam David Epqsuein Stepien G. Kobovrov Michael T . Goodsich Sergfy Pupyrev

. Circle Packing Md. Jawaherul Alam Sizes of circles may vary exponentially Any planar graph has a circle-packing [Koebe, 1936] Edges are contacts between circles Vertices are interior-disjoint circles GD 2014 9 9 8 5 8 4 2 1 7 3 5 6 4 3 1 2 ■ Contact representation with circles

. Circle Packing Md. Jawaherul Alam Sizes of circles may vary exponentially Any planar graph has a circle-packing [Koebe, 1936] GD 2014 9 9 8 5 8 4 2 1 7 3 5 6 4 3 1 2 ■ Contact representation with circles ■ Vertices are interior-disjoint circles ■ Edges are contacts between circles

. Circle Packing Md. Jawaherul Alam Sizes of circles may vary exponentially GD 2014 9 9 8 5 8 4 2 1 7 3 5 6 4 3 1 2 ■ Contact representation with circles ■ Vertices are interior-disjoint circles ■ Edges are contacts between circles √ Any planar graph has a circle-packing [Koebe, 1936]

. Circle Packing Md. Jawaherul Alam GD 2014 9 9 8 5 8 4 2 1 7 3 5 6 4 3 1 2 ■ Contact representation with circles ■ Vertices are interior-disjoint circles ■ Edges are contacts between circles √ Any planar graph has a circle-packing [Koebe, 1936] × Sizes of circles may vary exponentially

. Circle Packing: Variation in Sizes Goal: Balanced Circle-Packing Polynomial ratio between maximum and minimum diameter Md. Jawaherul Alam GD 2014

. Circle Packing: Variation in Sizes Goal: Balanced Circle-Packing Polynomial ratio between maximum and minimum diameter Md. Jawaherul Alam GD 2014

. Circle Packing: Variation in Sizes Goal: Balanced Circle-Packing Md. Jawaherul Alam GD 2014 ■ Polynomial ratio between maximum and minimum diameter

. Related Work Circle Packing: [Brightwell and Scheinerman, 1993] . Balanced Circle Packing: It is NP-complete to test whether a graph admits contact representation with unit circles [Breu and Kirkpatrick, 1998] . Disk Intersection Graphs: In a realization with integer radii, radius of is sometimes necessary and always sufficient [McDiarmid and Müller, 2013] . Md. Jawaherul Alam GD 2014 ■ Any plane graph has a circle-packing [Koebe, 1936] . ■ Any 3-connected plane graph has a primal-dual circle packing

. Related Work Circle Packing: [Brightwell and Scheinerman, 1993] . Balanced Circle Packing: representation with unit circles [Breu and Kirkpatrick, 1998] . Disk Intersection Graphs: In a realization with integer radii, radius of is sometimes necessary and always sufficient [McDiarmid and Müller, 2013] . Md. Jawaherul Alam GD 2014 ■ 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

. Related Work Circle Packing: [Brightwell and Scheinerman, 1993] . Balanced Circle Packing: representation with unit circles [Breu and Kirkpatrick, 1998] . Disk Intersection Graphs: necessary and always sufficient [McDiarmid and Müller, 2013] . Md. Jawaherul Alam GD 2014 ■ 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 2 2 Θ( n ) is sometimes

. Our Result Md. Jawaherul Alam GD 2014 Balanced circle packing p m n o l √ trees. g e f j h i k c b d √ cactus graphs. a √ outerpaths.

. Our Result Md. Jawaherul Alam GD 2014 Balanced circle packing p m n o l √ trees. g e f j h i k c b d √ cactus graphs. a √ outerpaths. √ bounded degree and O (log n ) outerplanarity.

. Our Result Md. Jawaherul Alam GD 2014 Balanced circle packing p m n o l √ trees. g e f j h i k c b d √ cactus graphs. a √ outerpaths. √ bounded degree and O (log n ) outerplanarity. × bounded degree but linear outerplanarity. × bounded outerplanarity but linear degree.

. Our Result Md. Jawaherul Alam GD 2014 Balanced circle packing p m n o l √ trees. g e f j h i k c b d √ cactus graphs. a √ outerpaths. √ bounded degree and O (log n ) outerplanarity. × bounded degree but linear outerplanarity. × bounded outerplanarity but linear degree. √ bounded tree-depth.

. Our Result Md. Jawaherul Alam GD 2014 Balanced circle packing p m n o l √ trees. g e f j h i k c b d √ cactus graphs. a √ outerpaths. √ bounded degree and O (log n ) outerplanarity. × bounded degree but linear outerplanarity. × bounded outerplanarity but linear degree. √ bounded tree-depth.

. Balanced Packing for Trees Md. Jawaherul Alam GD 2014 p l m n o g e f j h i k b c d a

. Balanced Packing for Trees Md. Jawaherul Alam – length is (roughly) proportional to the number of leaves in subtree GD 2014 p n o l g k e f m j h i b d c p l m n o g e f j h i k a b c d a ■ Compute balanced square-contact representation

. Balanced Packing for Trees Md. Jawaherul Alam – length is (roughly) proportional to the number of leaves in subtree GD 2014 p n o l g k e f m j h i b d c p l m n o g e f j h i k a b c d a ■ Compute balanced square-contact representation

. Balanced Packing for Trees Md. Jawaherul Alam GD 2014 p p n o n o l l g k g k e f e f m j m j h i h i b d b c d c p l m n o g e f j h i k a a b c d a ■ Compute balanced square-contact representation ■ Draw Inscribing circles inside the squares

. Balanced Packing for Trees Md. Jawaherul Alam GD 2014 p p n o n o l l g k g k e f e f m j m j h i h i b d b c d c p l m n o g e f j h i k a a b c d a ■ Compute balanced square-contact representation ■ Draw Inscribing circles inside the squares ■ Translate downwards

. Balanced Packing for Trees Md. Jawaherul Alam GD 2014 p n o p l n o l g k e f m m j k f g e h i j h i b c d b c d p l m n o g e f j h i k a a b c d a ■ Compute balanced square-contact representation ■ Draw Inscribing circles inside the squares ■ Translate downwards

. Augmented Fan-Trees Md. Jawaherul Alam Any subgraph of an augmented fan-tree has a balanced packing Claim: GD 2014 p p m n o m n o l l g g e f j e f j h i k h i k c c b d b d a a ■ Add a path between the children of every vertex

. Augmented Fan-Trees Md. Jawaherul Alam Any subgraph of an augmented fan-tree has a balanced packing Claim: GD 2014 p p m n o m n o l l g g e f j e f j h i k h i k c c b d b d a a ■ Add a path between the children of every vertex

. Packing for Subgraphs of Augmented Fan-Trees Md. Jawaherul Alam GD 2014 1 2 3 4 5 6 7 1 2 3 4 5 6 7 p p 3 5 1 4 3 4 5 2 6 2 6 1 7 7 ε p p ■ Follow the algorithm for balanced packing of the tree ■ Modify the circles for the children of each vertex

. Balanced Packing for Cactus Graphs Md. Jawaherul Alam GD 2014

. Balanced Packing for Cactus Graphs Md. Jawaherul Alam GD 2014

. Balanced Packing for Cactus Graphs Each biconnected component is a cycle or a single edge Each cactus graph is a subgraph of an augmented fan-tree Md. Jawaherul Alam GD 2014

. Balanced Packing for Cactus Graphs Each biconnected component is a cycle or a single edge Md. Jawaherul Alam GD 2014 ■ Each cactus graph is a subgraph of an augmented fan-tree

. Balanced Packing for Cactus Graphs Each biconnected component is a cycle or a single edge Each cactus graph admits a balanced packing Md. Jawaherul Alam GD 2014 ■ Each cactus graph is a subgraph of an augmented fan-tree

. Balanced Packing for Cactus Graphs Each biconnected component is a cycle or a single edge Md. Jawaherul Alam GD 2014 ■ Each cactus graph is a subgraph of an augmented fan-tree ⇒ Each cactus graph admits a balanced packing

. Balanced Packing for Outerpaths Outerplanar graph whose weak dual is a path Md. Jawaherul Alam GD 2014

. Balanced Packing for Outerpaths Md. Jawaherul Alam Outerplanar graph whose weak dual is a path GD 2014 3 2 0 0 1 2 3 4 5 6 7 8 9 ■ Draw Circles for spine vertices

. Balanced Packing for Outerpaths Md. Jawaherul Alam Outerplanar graph whose weak dual is a path GD 2014 θ θ 3 2 0 0 1 2 3 4 5 6 7 8 9 θ θ ■ Draw Circles for spine vertices ■ Rotate to create space for other vertices

. Our Result Balanced circle packing Md. Jawaherul Alam GD 2014 √ trees. √ cactus graphs. √ outerpaths. √ bounded degree and O (log n ) outerplanarity. × bounded degree but linear outerplanarity. × bounded outerplanarity but linear degree. √ bounded tree-depth.

. Our Result Balanced circle packing Md. Jawaherul Alam GD 2014 √ trees. √ cactus graphs. √ outerpaths. √ bounded degree and O (log n ) outerplanarity. × bounded degree but linear outerplanarity. × bounded outerplanarity but linear degree. √ bounded tree-depth.