Recognizing and Drawing IC-planar Graphs Philipp Kindermann - PowerPoint PPT Presentation

Triangulation + Matching Given a triconnected plane graph T = ( V, E T ) and a matching M = ( V, E M ), is G = ( V, E T ∪ E M ) IC-planar? u r uv l uv v

Triangulation + Matching Given a triconnected plane graph T = ( V, E T ) and a matching M = ( V, E M ), is G = ( V, E T ∪ E M ) IC-planar? Interior I ( u, v ) u r uv l uv v

Triangulation + Matching Given a triconnected plane graph T = ( V, E T ) and a matching M = ( V, E M ), is G = ( V, E T ∪ E M ) IC-planar? The boundaries of two Interior I ( u, v ) interiors may not intersect. u r uv l uv v

Triangulation + Matching Given a triconnected plane graph T = ( V, E T ) and a matching M = ( V, E M ), is G = ( V, E T ∪ E M ) IC-planar? The boundaries of two Interior I ( u, v ) interiors may not intersect. u � r uv l uv v

Triangulation + Matching Given a triconnected plane graph T = ( V, E T ) and a matching M = ( V, E M ), is G = ( V, E T ∪ E M ) IC-planar? The boundaries of two Interior I ( u, v ) interiors may not intersect. u � r uv l uv � v

Triangulation + Matching Given a triconnected plane graph T = ( V, E T ) and a matching M = ( V, E M ), is G = ( V, E T ∪ E M ) IC-planar? The boundaries of two Interior I ( u, v ) interiors may not intersect. u � � r uv l uv � v

Triangulation + Matching Given a triconnected plane graph T = ( V, E T ) and a matching M = ( V, E M ), is G = ( V, E T ∪ E M ) IC-planar? The boundaries of two Interior I ( u, v ) interiors may not intersect. u � � r uv l uv × � v

Triangulation + Matching

Triangulation + Matching u v

Triangulation + Matching u v

Triangulation + Matching u v

Triangulation + Matching u a v b

Triangulation + Matching u a v b

Triangulation + Matching u a v b

Triangulation + Matching u a c d v b

Triangulation + Matching u a c d v b

Triangulation + Matching u a c d v b

Triangulation + Matching H : u a c d v b Hierarchical structure: Tree H = ( V H , E H )

Triangulation + Matching H : u a I uv c d v I ab I cd b Hierarchical structure: Tree H = ( V H , E H ) V H = {I uv | ( u, v ) ∈ M }

Triangulation + Matching H : G u a I uv c d v I ab I cd b Hierarchical structure: Tree H = ( V H , E H ) V H = {I uv | ( u, v ) ∈ M } ∪ { G }

Triangulation + Matching H : G u a I uv c d v I ab I cd b Hierarchical structure: Tree H = ( V H , E H ) V H = {I uv | ( u, v ) ∈ M } ∪ { G } ( I uv , I ab ) ∈ E H ⇔ I uv ⊂ I ab

Triangulation + Matching H : G u a I uv c d v I ab I cd b Hierarchical structure: Tree H = ( V H , E H ) V H = {I uv | ( u, v ) ∈ M } ∪ { G } ( I uv , I ab ) ∈ E H ⇔ I uv ⊂ I ab outdeg( I uv ) = 0 ⇒ ( I uv , G ) ∈ E H

Triangulation + Matching G u a I uv c d v I ab I cd b

Triangulation + Matching G a I uv c d I ab I cd b

Triangulation + Matching G a I uv c d I ab I cd b

Triangulation + Matching G a I uv c d I ab I cd b

Triangulation + Matching G a I uv c d I ab I cd b

Triangulation + Matching G a I uv c d I ab I cd b

Triangulation + Matching G a I uv c d I ab I cd b

Triangulation + Matching G a I uv c d I ab I cd b • Always pick “middle” routing

Triangulation + Matching G a I uv c d I ab I cd b • Always pick “middle” routing • Solve rest with 2SAT

Triangulation + Matching G u a I uv c d v I ab I cd b • Always pick “middle” routing • Solve rest with 2SAT

Triangulation + Matching G u a I uv c d v I ab I cd b • Always pick “middle” routing • Solve rest with 2SAT • Recursively check which routings are valid

Triangulation + Matching G u a I uv c d v I ab I cd b • Always pick “middle” routing • Solve rest with 2SAT • Recursively check which routings are valid

Triangulation + Matching G u a I uv c d v I ab I cd b • Always pick “middle” routing • Solve rest with 2SAT • Recursively check which routings are valid

Triangulation + Matching G u a I uv c d v I ab I cd b • Always pick “middle” routing • Solve rest with 2SAT • Recursively check which routings are valid

Triangulation + Matching Theorem. G IC-planarity can be tested efficiently if the input graph is a u a triangulated planar graph and a matching I uv c d v I ab I cd b • Always pick “middle” routing • Solve rest with 2SAT • Recursively check which routings are valid

Straight-Line Drawings Theorem. IC-plane graphs can be drawn straight-line on the O ( n ) × O ( n ) grid in O ( n ) time.

Straight-Line Drawings Theorem. IC-plane graphs can be drawn straight-line on the O ( n ) × O ( n ) grid in O ( n ) time. Using a special 1-planar drawing... [Alam et al. GD’13]

Straight-Line Drawings Theorem. RAC? IC-plane graphs can be drawn straight-line on the O ( n ) × O ( n ) grid in O ( n ) time. Using a special 1-planar drawing... [Alam et al. GD’13]

Straight-Line Drawings Theorem. RAC? IC-plane graphs can be drawn straight-line on the O ( n ) × O ( n ) grid in O ( n ) time. Using a special 1-planar drawing... [Alam et al. GD’13]

Straight-Line Drawings Theorem. RAC? IC-plane graphs can be drawn straight-line on the O ( n ) × O ( n ) grid in O ( n ) time. Using a special 1-planar drawing... [Alam et al. GD’13] Theorem. Straight-line RAC drawings of IC-planar graphs may require exponential area.

Straight-Line Drawings Theorem. RAC? IC-plane graphs can be drawn straight-line on the O ( n ) × O ( n ) grid in O ( n ) time. Using a special 1-planar drawing... [Alam et al. GD’13] Theorem. Straight-line RAC drawings of IC-planar graphs may require exponential area.

Straight-Line RAC Drawings Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90]

Straight-Line RAC Drawings Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] • Augment to 3-connected planar graph

Straight-Line RAC Drawings Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] • Augment to 3-connected planar graph • Insert vertices in canonical order

Straight-Line RAC Drawings Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] • Augment to 3-connected planar graph • Insert vertices in canonical order • Contour only has slopes ± 1

Straight-Line RAC Drawings Adjust Shift-Algorithm for planar graphs [de Fraysseix, Pach & Pollack Comb’90] • Augment to 3-connected planar graph • Insert vertices in canonical order • Contour only has slopes ± 1