Strongly Monotone Drawings of Planar Graphs Stefan Felsner - - PowerPoint PPT Presentation
Strongly Monotone Drawings of Planar Graphs Stefan Felsner Technische Universit at Berlin FernUniversit at in Hagen Alexander Igamberdiev Philipp Kindermann FernUniversit at in Hagen Freie Universit at Berlin Boris Klemz
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
EuroCG 2016, Lugano Stefan Felsner
Graz University of Technology Alexander Igamberdiev Philipp Kindermann Boris Klemz Tamara Mchedlidze Manfred Scheucher Karlsruhe Institute of Technology Freie Universit¨ at Berlin FernUniversit¨ at in Hagen FernUniversit¨ at in Hagen Technische Universit¨ at Berlin
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
We want planar straight-line graph drawings that are ... Strongly monotone: between each vertex pair (u, v) exists a path that is monotonically increasing in direction d = − → uv u v strongly monotone d
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
We want planar straight-line graph drawings that are ... Strongly monotone: between each vertex pair (u, v) exists a path that is monotonically increasing in direction d = − → uv u v strongly monotone u v not strongly monotone w w d d
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
We want planar straight-line graph drawings that are ... Strongly monotone: between each vertex pair (u, v) exists a path that is monotonically increasing in direction d = − → uv u v strongly monotone u v not strongly monotone Motivation: Path-finding tasks become easy! w w d d
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Strongly monotone drawings: do not exist for every planar graph
[Kindermann et al. ’14]
exist for every tree
[Kindermann et al. ’14]
exist for every 2-connected outerplanar graph
[Kindermann et al. ’14]
area required can be exponential
[N¨
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Strongly monotone drawings: do not exist for every planar graph
[Kindermann et al. ’14]
exist for every tree
[Kindermann et al. ’14]
exist for every 2-connected outerplanar graph
[Kindermann et al. ’14]
area required can be exponential
[N¨
exist for every planar 3-connectd graph exist for every outerplanar graph exist for every 2-tree
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Theorem: Every 3-connected planar graph has a strongly monotone drawing. Proof idea: Every 3-connected planar graph G admits a primal-dual circle packing P. [Brightwell, Scheinerman 1993] Drawing induced by P is strongly monotone.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
primal dual circle packing P of G
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
circle contact representations of ... (primal) graph G primal dual circle packing P of G
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
circle contact representations of ... (primal) graph G dual graph of G primal dual circle packing P of G
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe
face circle = inscribed circle of face
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
circle contact representations of ... (primal) graph G dual graph of G ... which are orthogonal: edge e crosses its dual edge e∗ perpendicularly in edge point pe primal dual circle packing P of G pe vertex v has ’star-shaped’ Region Rv v can see all of Rv!
face circle = inscribed circle of face
v
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Consider vertices v1, vk where w.l.o.g. s = v1vk horizontal. v1 vk
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
f1 f2 f3 f4 R1 R2 R3 R4 Rk General position: s does not pass through circle centers or edge points. Consider vertices v1, vk where w.l.o.g. s = v1vk horizontal. v1 vk ⇒ s intersects alternating sequence of vertex regions and face circles:
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
f1 f2 f3 f4 R1 R2 R3 R4 Rk General position: s does not pass through circle centers or edge points. Concatenation yields strongly monotone (v1, vk)-path! Consider vertices v1, vk where w.l.o.g. s = v1vk horizontal. v1 vk ⇒ s intersects alternating sequence of vertex regions and face circles: Each region Ri has some vertex vi in its center. Idea: Find path Pi from vi of Ri to vi+1 of Ri+1, 1 ≤ i < k.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) ci x
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci x
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci x
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci x
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er x
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. x Edge points p1, . . . , pr are above s: p1 p2 pr
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x Edge points p1, . . . , pr are above s: p1 p2 pr
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x Same for edge point of er. Edge points p1, . . . , pr are above s: p1 p2 pr
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x ⇒ all edge points p1, . . . pr above ci. p1, . . . , pr in this order around fi. Same for edge point of er. Edge points p1, . . . , pr are above s: p1 p2 pr
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
s Ri vi vi+1 Ri+1 fi construction of Pi = (e1, . . . , er) Circle fi is inscribed circle of inner face ⇒ two paths leading from vi to vi+1 ci If ci is below s, pick the upper path. Otherwise the lower path. e1 e2 er e1, . . . , er tangent to fi. Edge point of e1 is above s ⇐ vi sees x and e1 points upwards. x ⇒ all edge points p1, . . . pr above ci. p1, . . . , pr in this order around fi.
⇒ strongly monotone Same for edge point of er. Edge points p1, . . . , pr are above s: p1 p2 pr
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently:
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: an expanded edge
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: an expanded edge
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: an expanded edge
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Graph generated from a K3 by stacking vertices on edges. Graph generated from an edge by multistacking once per edge. Equivalently: Theorem: Every 2-tree has a strongly monotone drawing. an expanded edge
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
A drawing with bubbles (DWB) is a straight-line drawing of G = (V, E) where every edge in some E′ ⊆ E is associated with a bubble.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
A drawing with bubbles (DWB) is a straight-line drawing of G = (V, E) where every edge in some E′ ⊆ E is associated with a bubble. An extension of a DWB is a drawing obtained by stacking one vertex into each bubble of some E′′ ⊆ E′.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
A drawing with bubbles (DWB) is a straight-line drawing of G = (V, E) where every edge in some E′ ⊆ E is associated with a bubble. An extension of a DWB is a drawing obtained by stacking one vertex into each bubble of some E′′ ⊆ E′. A DWB is strongly monotone if every extension is strongly monotone. i.e. the drawing without bubbles is strongly monotone stacking into bubbles maintains strong monotonicty
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Base case: e
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step:
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge e B
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u e e B C u
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble e e B C u
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble e e B C u
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble e e B C u
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Idea: Draw according to multistacking sequence. Invariant: Before / after each multistack ... every non-expanded edge has a bubble the DWB is strongly monotone stacking gives obtuse angles Induction step: next edge place vertices on circular arc C centered at u New DWB is strongly monotone: paths between 2 new vertices paths between new vertex and old vertex / bubble How to add bubbles for new edges? e e B C u Not done yet!
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α Idea: Find α s.t. between any vertex / bubble pair exists an α-safe strongly monotone path.
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α Idea: Find α s.t. between any vertex / bubble pair exists an α-safe strongly monotone path. Place bubble for e = (u, v) close to v s.t. prolonging paths leading to v changes the direction by < α. v u w
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
A strongly monotone path P is α-safe if it remains monotone while the direction is tilted by at most α. α α Idea: Find α s.t. between any vertex / bubble pair exists an α-safe strongly monotone path. Place bubble for e = (u, v) close to v s.t. prolonging paths leading to v changes the direction by < α. v u v u x w w
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Strongly monotone drawings: do not exist for every planar graph
[Kindermann et al. ’14]
exist for every tree
[Kindermann et al. ’14]
exist for every 2-connected outerplanar graph
[Kindermann et al. ’14]
area required can be exponential
[N¨
exist for every planar 3-connectd graph exist for every outerplanar graph exist for every 2-tree
exist for every planar 2-connected graph?
Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs
Strongly monotone drawings: do not exist for every planar graph
[Kindermann et al. ’14]
exist for every tree
[Kindermann et al. ’14]
exist for every 2-connected outerplanar graph
[Kindermann et al. ’14]
area required can be exponential
[N¨
exist for every planar 3-connectd graph exist for every outerplanar graph exist for every 2-tree
exist for every planar 2-connected graph?
Thank you!