Smooth Orthogonal Drawings of Planar Graphs Philipp Kindermann - PowerPoint PPT Presentation

SC 2 -Layouts biconnected 4 -planar graph → orthogonal complexity-2 layout smooth → → → → n 1 3 2

SC 2 -Layouts biconnected 4 -planar graph → orthogonal complexity-2 layout smooth → → → → n 1 3 2

SC 2 -Layouts biconnected 4 -planar graph → orthogonal complexity-2 layout smooth → → → → → n 1 3 2

SC 2 -Layouts biconnected 4 -planar graph → orthogonal complexity-2 layout smooth → → → → → n 1 3 2

SC 2 -Layouts biconnected 4 -planar graph → orthogonal complexity-2 layout smooth → → → → → → n 1 3 2

SC 2 -Layouts biconnected 4 -planar graph → orthogonal complexity-2 layout smooth → → → → → → n 1 3 2

Crossings

Crossings

Crossings

Crossings

Crossings

Crossings

Cut Def.

Cut • y -monotone curve Def. u

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments u u

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments • divides the current drawing into a left and a right part u u

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments • divides the current drawing into a left and a right part • intersects only horizontal segments u u u

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments • divides the current drawing into a left and a right part • intersects only horizontal segments u u u u

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments • divides the current drawing into a left and a right part • intersects only horizontal segments u u u u u

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments • divides the current drawing into a left and a right part • intersects only horizontal segments u u u u u Problems:

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments • divides the current drawing into a left and a right part • intersects only horizontal segments u u u u u Problems:

Cut • y -monotone curve Def. • consists of horizontal, vertical and circular segments • divides the current drawing into a left and a right part • intersects only horizontal segments u u u u u Problems:

Invariants ( I 1 ) Every open edge is associated with a column

Invariants ( I 1 ) Every open edge is associated with a column An L-shape always contains a horizontal segment; ( I 2 ) it never contains a vertical segment.

Invariants ( I 1 ) Every open edge is associated with a column An L-shape always contains a horizontal segment; ( I 2 ) it never contains a vertical segment. ( I 3 ) A C-shape always has a horizontal segment incident to its bottom vertex.

Maintain invariants L-shape

Maintain invariants L-shape C-shape

Maintain invariants L-shape Double C-shape C-shape

Maintain invariants L-shape Double C-shape C-shape protected

Invariants, updated ( I 1 ) Every open edge is associated with a column An L-shape always contains a horizontal segment; ( I 2 ) it never contains a vertical segment. ( I 3 ) An unprotected C-shape always has a horizontal segment incident to its bottom vertex.

Eliminate crossings v u v u

Eliminate crossings 1. move v up v u v u

Eliminate crossings 1. move v up v u v u

Eliminate crossings 1. move v up v 2. find a cut u v u

Eliminate crossings 1. move v up v 2. find a cut u v u

Eliminate crossings 1. move v up v 2. find a cut 3. move vertices to the left u v u

Eliminate crossings 1. move v up v 2. find a cut 3. move vertices to the left u v u

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1

Example Run 1