1 Visualization of Graphs Lecture 11: Beyond Planarity Drawing Graphs with Crossings Part I: Graph Classes and Drawing Styles Jonathan Klawitter Partially based on slides by Fabrizio Montecchini, Michalis Bekos, and Walter Didimo. 2 - 9 Planar Graphs Planar graphs admit drawings in the plane without crossings. Plane graph is a planar graph with a plane embedding = rotation system. Planarity is recognizable in linear time. Different drawing styles... 6 6 6 6 4 5 4 5 4 5 4 5 1 3 1 3 2 3 2 1 3 2 1 2 straight-line drawing orthogonal drawing grid drawing with circular-arc drawing bends & 3 slopes 3 - 7 And Non-Planar Graphs? We have seen a few drawing styles: force-directed drawing hierarchical drawing orthogonal layouts (via planarization) Maybe not all crossings are equally bad? 1 2 3 4 5 6 7 block crossings Which crossings feel worse? 4 - 9 Eye-Tracking Experiment [Eades, Hong & Huang 2008] Input: A graph drawing and designated path. Task: Trace path and count number of edges. Results: no crossings eye movements smooth and fast large crossing angles eye movements smooth but slightly slower small crossing angles eye movements no longer smooth and very slow (back-and-forth movements at crossing points) 5 - 10 Some Beyond-Planar Graph Classes We define aesthetics for edge crossings and avoid/minimize \bad" crossing configurations. k-planar (k = 1) k-quasi-planar (k = 3) fan-planar RAC right-angle crossing X X X X topological graphs geometric graphs 5 - 15 Some Beyond-Planar Graph Classes We define aesthetics for edge crossings and avoid/minimize \bad" crossing configurations. k-planar (k = 1) k-quasi-planar (k = 3) fan-planar RAC right-angle crossing X X X X There are many more beyond planar graph classes. combinations, . X X X IC (independent crossing) fan-crossing-free skewness-k (k = 2) 6 - 8 Drawing Styles for Crossings RAC orthogonal slanted orthogonal block/bundle crossings right-angle crossing circular layout: 28 invididual vs. 12 bundle crossings X sym. partial cased crossings 1/4-SHPED edge drawing 7 - 13 Geometric Representations c K5 c d e d e b a a b bar 1-visibility thickness rectangle visibility representation (B1VR) two graph representation Every 1-planar graph admits a B1VR. G has at most 6n − 20 edges [Bose et al. 1997] [Brandenburg 2014; Evans et al. 2014; [Shermer 1996] Angelini et al. 2018] Recognition is NP-complete Recognition becomes polynomial if embedding is fixed [Biedl et al. 2018] 8 GD Beyond Planarity: a Taxonomy GD Beyond Planarity Density Recognition Stretchability Relationships Constraints Eng. & Exper. many 1-plan. - RAC &. 1-plan. circ. & layers families simult. - fan-plan. & k-plan. - 2-layer fan & RAC - k-plan. & k-qu.-plan. fixed rot. variable - qu.-pl. system embedding - circ. RAC - out. 1-plan. - RAC - 2-layer RAC Aesthetics - 2-layer fan NP-hard poly-time NP-hard poly-time - book emb. area edge-complexity - 1-plan. - out. 1-plan. - RAC maximal - RAC - 2-layer RAC - fan-plan. 1-plan. - 1-plan. - out. fan-plan. - k-plan. - fan-plan. - opt. 1-plan. slopes - 2-layer RAC - RAC bends - book emb. - 1-plan. - 2-layer fan-plan - quas.-plan. - k-plan. out. 1-plan. - RAC Taken from: G. Liotta, Invited talk at SoCG 2017 "Graph Drawing Beyond Planarity: Some Results and Open Problems", Jul. 2017 9 Visualization of Graphs Lecture 11: Beyond Planarity Drawing Graphs with Crossings Part II: Density & Relationships Jonathan Klawitter 10 GD Beyond Planarity: a Taxonomy GD Beyond Planarity Density Recognition Stretchability Relationships Constraints Eng. & Exper. many 1-plan. - RAC &. 1-plan. circ. & layers families simult. - fan-plan. & k-plan. - 2-layer fan & RAC - k-plan. & k-qu.-plan. fixed rot. variable - qu.-pl. system embedding - circ. RAC - out. 1-plan. - RAC - 2-layer RAC Aesthetics - 2-layer fan NP-hard poly-time NP-hard poly-time - book emb. area edge-complexity - 1-plan. - out. 1-plan. - RAC maximal - RAC - 2-layer RAC - fan-plan. 1-plan. - 1-plan. - out. fan-plan. - k-plan. - fan-plan. - opt. 1-plan. slopes - 2-layer RAC - RAC bends - book emb. - 1-plan. - 2-layer fan-plan - quas.-plan. - k-plan. out. 1-plan. - RAC Taken from: G. Liotta, Invited talk at SoCG 2017 "Graph Drawing Beyond Planarity: Some Results and Open Problems", Jul. 2017 11 - 5 Density of 1-Planar Graphs Theorem. [Ringel 1965, Pach & T´oth1997] A 1-planar graph with n vertices has at most4 n − 8 edges, which is a tight bound. Proof sketch. red edges do not cross each blue edge crosses a green edge 11 - 7 Density of 1-Planar Graphs Theorem. [Ringel 1965, Pach & T´oth1997] A 1-planar graph with n vertices has at most4 n − 8 edges, which is a tight bound. Proof sketch. red edges do not cross each blue edge crosses a green edge red-blue plane graph Grb Grb mrb ≤ 3n − 6 11 - 9 Density of 1-Planar Graphs Theorem. [Ringel 1965, Pach & T´oth1997] A 1-planar graph with n vertices has at most4 n − 8 edges, which is a tight bound. Proof sketch. red edges do not cross each blue edge crosses a green edge red-blue plane graph Grb Gg mrb ≤ 3n − 6 green plane graph Gg mg ≤ 3n − 6 11 - 14 Density of 1-Planar Graphs Theorem. [Ringel 1965, Pach & T´oth1997] A 1-planar graph with n vertices has at most4 n − 8 edges, which is a tight bound. Proof sketch. red edges do not cross each blue edge crosses a green edge red-blue plane graph Grb mrb ≤ 3n − 6 green plane graph Gg mg ≤ 3n − 6 ) m ≤ mrb + mg ≤ 6n − 12 Observe that each green edge joins two faces in Grb. mg ≤ frb=2 ≤ (2n − 4)=2 = n − 2 11 - 17 Density of 1-Planar Graphs Theorem. [Ringel 1965, Pach & T´oth1997] A 1-planar graph with n vertices has at most4 n − 8 edges, which is a tight bound. Proof sketch. red edges do not cross each blue edge crosses a green edge red-blue plane graph Grb Planar structure: mrb ≤ 3n − 6 2n − 4 edges green plane graph Gg n − 2 faces mg ≤ 3n − 6 ) m ≤ mrb + mg ≤ 6n − 12 Observe that each green edge joins two faces in Grb. mg ≤ frb=2 ≤ (2n − 4)=2 = n − 2 ) m = mrb + mg ≤ 3n − 6 + n − 2 = 4n − 8 11 - 19 Density of 1-Planar Graphs Theorem. [Ringel 1965, Pach & T´oth1997] A 1-planar graph with n vertices has at most4 n − 8 edges, which is a tight bound. Proof sketch. red edges do not cross each blue edge crosses a green edge red-blue plane graph Grb Planar structure: mrb ≤ 3n − 6 2n − 4 edges green plane graph Gg n − 2 faces mg ≤ 3n − 6 ) m ≤ mrb + mg ≤ 6n − 12 Edges per face: 2 edges Total: 4n − 8 edges Observe that each green edge joins two faces in Grb. mg ≤ frb=2 ≤ (2n − 4)=2 = n − 2 ) m = mrb + mg ≤ 3n − 6 + n − 2 = 4n − 8 12 - 9 Density of 1-Planar Graphs Theorem. [Ringel 1965, Pach & T´oth1997] A 1-planar graph with n vertices has at most4 n − 8 edges, which is a tight bound. A 1-planar graph with n vertices is called optimal if it has exactly 4n − 8 edges. n = 12; m = 40 A 1-planar graph is called maximal if adding any edge would result in a non-1-planar graph. Theorem. [Brandenburg et al. 2013] There are maximal 1-planar graphs with n vertices and 45=17n − O(1) edges. ≈ 2:65n Theorem. [Didimo 2013] n = 20; m = 48 A 1-planar graph with n vertices that admits a straight-line drawing has at most4 n − 9 edges. 13 - 9 Density of k-Planar Graphs Theorem. A k-planar graph with n vertices has at most: k number of edges 0 3(n − 2) Euler's formula 1 4(n − 2) [Ringel 1965] 2 5(n − 2) [Pach and T´oth1997] optimal 2-planar Planar structure: n − m + f = 2 Edges per face: m = c · f ? Total: 13 - 13 Density of k-Planar Graphs Theorem. A k-planar graph with n vertices has at most: k number of edges 0 3(n − 2) Euler's formula 1 4(n − 2) [Ringel 1965] 2 5(n − 2) [Pach and T´oth1997] optimal 2-planar Planar structure: 5 3 (n − 2) edges 2 3 (n − 2) faces n − m + f = 2 Edges per face: 5edges m = c · f ? Total: 5(n − 2) edges 5 m = 2 f 13 - 16 Density of k-Planar Graphs Theorem. A k-planar graph with n vertices has at most: 7 8 k number of edges 9 1 2 0 3(n − 2) Euler's formula 5 3 4 6 1 4(n − 2) [Ringel 1965] 2 5(n − 2) [Pach and T´oth1997] optimal 3-planar 3 5:5(n − 2) [Pach et al. 2006] 13 - 17 Density of k-Planar Graphs Theorem. A k-planar graph with n vertices has at most: k number of edges 7 8 1 2 0 3(n − 2) Euler's formula 5 3 4 6 1 4(n − 2) [Ringel 1965] 2 5(n − 2) [Pach and T´oth1997] optimal 3-planar 3 5:5(n − 2) [Pach et al. 2006] 13 - 20 Density of k-Planar Graphs Theorem. A k-planar graph with n vertices has at most: k number of edges 0 3(n − 2) Euler's formula 1 4(n − 2) [Ringel 1965] 2 5(n − 2) [Pach and T´oth1997] optimal 3-planar 3 5:5(n − 2) [Pach et al.