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 Karlsruhe Institute of Technology Tamara Mchedlidze Graz University of Technology Manfred Scheucher EuroCG 2016, Lugano Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Strongly Monotone Drawings 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 d v u strongly monotone Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Strongly Monotone Drawings 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 d d v v u u w w strongly monotone not strongly monotone Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Strongly Monotone Drawings 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 d d v v u u w w strongly monotone not strongly monotone Motivation: Path-finding tasks become easy! Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Related Work & Results Strongly monotone drawings: do not exist for every planar graph [Kindermann et al. ’14] exist for every 2-connected outerplanar graph [Kindermann et al. ’14] exist for every tree [Kindermann et al. ’14] area required can be exponential [N¨ ollenburg et al. ’14] Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Related Work & Results Strongly monotone drawings: do not exist for every planar graph [Kindermann et al. ’14] exist for every 2-connected outerplanar graph [Kindermann et al. ’14] exist for every tree [Kindermann et al. ’14] area required can be exponential [N¨ ollenburg et al. ’14] } exist for every planar 3-connectd graph exist for every outerplanar graph our results exist for every 2-tree Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

3-connected 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 Packings primal dual circle packing P of G Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings primal dual circle packing P of G circle contact representations of ... ( primal ) graph G Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings primal dual circle packing P of G circle contact representations of ... ( primal ) graph G dual graph of G Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings primal dual circle packing P of G circle contact representations of ... ( primal ) graph G dual graph of G ... which are orthogonal : edge e crosses its dual edge e ∗ p e perpendicularly in edge point p e Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings primal dual circle packing P of G circle contact representations of ... ( primal ) graph G dual graph of G ... which are orthogonal : edge e crosses its dual edge e ∗ p e perpendicularly in edge point p e Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings primal dual circle packing P of G circle contact representations of ... ( primal ) graph G dual graph of G ... which are orthogonal : edge e crosses its dual edge e ∗ p e perpendicularly in edge point p e Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings primal dual circle packing P of G circle contact representations of ... ( primal ) graph G dual graph of G ... which are orthogonal : face circle = inscribed circle of face ⇒ edge e crosses its dual edge e ∗ p e perpendicularly in edge point p e Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Primal Dual Circle Packings primal dual circle packing P of G circle contact representations of ... ( primal ) graph G dual graph of G ... which are orthogonal : face circle = inscribed circle of face ⇒ ⇒ vertex v has ’star-shaped’ Region R v edge e crosses its dual edge e ∗ p e perpendicularly v v can see all of R v ! in edge point p e Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Constructing Strongly Monotone Paths Consider vertices v 1 , v k where w.l.o.g. s = v 1 v k horizontal. v 1 v k Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Constructing Strongly Monotone Paths Consider vertices v 1 , v k where w.l.o.g. s = v 1 v k horizontal. v 1 v k General position : s does not pass through circle centers or edge points. ⇒ s intersects alternating sequence of vertex regions and face circles: R 1 f 1 R 2 f 2 R 3 f 3 R 4 f 4 R k Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Constructing Strongly Monotone Paths Consider vertices v 1 , v k where w.l.o.g. s = v 1 v k horizontal. v 1 v k General position : s does not pass through circle centers or edge points. ⇒ s intersects alternating sequence of vertex regions and face circles: R 1 f 1 R 2 f 2 R 3 f 3 R 4 f 4 R k Each region R i has some vertex v i in its center. Idea: Find path P i from v i of R i to v i +1 of R i +1 , 1 ≤ i < k . Concatenation yields strongly monotone ( v 1 , v k )-path! Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle R i v i s x v i +1 c i R i +1 f i construction of P i = ( e 1 , . . . , e r ) Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle Circle f i is inscribed circle of inner face ⇒ two paths leading from v i to v i +1 R i v i s x v i +1 c i R i +1 f i construction of P i = ( e 1 , . . . , e r ) Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle Circle f i is inscribed circle of inner face ⇒ two paths leading from v i to v i +1 R i v i s x v i +1 c i R i +1 f i construction of P i = ( e 1 , . . . , e r ) Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle Circle f i is inscribed circle of inner face ⇒ two paths leading from v i to v i +1 R i v i s x v i +1 c i R i +1 f i construction of P i = ( e 1 , . . . , e r ) Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle Circle f i is inscribed circle of inner face ⇒ two paths leading from v i to v i +1 e 2 e 1 If c i is below s , pick the upper path. R i v i e r s Otherwise the lower path. x v i +1 c i R i +1 f i construction of P i = ( e 1 , . . . , e r ) Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle Circle f i is inscribed circle of inner face ⇒ two paths leading from v i to v i +1 p 2 e 2 p 1 p r e 1 If c i is below s , pick the upper path. R i v i e r s Otherwise the lower path. x v i +1 c i e 1 , . . . , e r tangent to f i . R i +1 f i Edge points p 1 , . . . , p r are above s : construction of P i = ( e 1 , . . . , e r ) Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs

Walking around a Face Circle Circle f i is inscribed circle of inner face ⇒ two paths leading from v i to v i +1 p 2 e 2 p 1 p r e 1 If c i is below s , pick the upper path. R i v i e r s Otherwise the lower path. x v i +1 c i e 1 , . . . , e r tangent to f i . R i +1 f i Edge points p 1 , . . . , p r are above s : Edge point of e 1 is above s ⇐ v i sees x and e 1 points upwards. construction of P i = ( e 1 , . . . , e r ) Felsner, Igamberdiev, Kindermann, Klemz, Mchedlidze, Scheucher Strongly Monotone Drawings of Planar Graphs