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INF562, Lecture 4: Geometric and Combinatorial Properties of Planar Graphs

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INF562, Lecture 4: Geometric and Combinatorial Properties of Planar Graphs INF562, Lecture 4: Geometric and combinatorial properties of planar graphs 5 jan 2016 Luca Castelli Aleardi 1 Intro Graph drawing: motivations and applications 2 Graph drawing and data visualization Global transportation system 3 Graph drawing and data visualization Roads, railways, ... 4 Graph drawing and data visualization Social network graph Community detection (via graph clustering) 5 Planar graphs french roads network Design of integrated circuits (VLSI) 6 Planar graphs french roads network Design of integrated circuits (VLSI) www.2m40.com 9 accidents en 2012 (last one, on 28th sptember) 6 Meshes and graphs in computational geometry Delaunay triangulations, Voronoi diagrams, planar meshes, ... GIS Technology triangles meshes already used in early 19th century (Delambre et Mchain) Delaunay triangulation Terrain modelling Planar mesh by L. Rineau, M. Yvinec Spherical Parameterization (Sheffer Gotsman) Voronoi diagram 7 Mesh parameterization (and straight line drawing) 2 given a (planar) graph G compute a mapping ρ :(VG) −! R s.t. edges are straight line segments without crossings 2 (given a 3D mesh M compute a bijective mapping ρ :(VG) −! R ) ρ(vi) = (xi; yi) 8 Mesh parameterization: applications Texture mapping and morphing (pictures by A. Sheffer) 9 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 10 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 adjacency matrix 1 if vi is adjacent to vj AG[i; j] = f 0 otherwise 10 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 10 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 10 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 10 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 10 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 10 Graph drawing: motivation 0 1 1 0 1 1 Challenge: what kind of graph does AG represent? 1 0 1 1 0 1 1 1 0 1 1 0 AG = 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 10 Part I Major results in graph theory 11 Major results (on planar graphs) in graph theory 12 Major results (on planar graphs) in graph theory Kuratowski theorem (1930) (cfr Wagner’s theorem, 1937) • G contains neither K5 nor K3;3 as minors 12 Major results (on planar graphs) in graph theory Kuratowski theorem (1930) (cfr Wagner’s theorem, 1937) • G contains neither K5 nor K3;3 as minors F´arytheorem (1947) • Every (simple) planar graph admits a straight line planar embedding (no edge crossings) v1 v1 v1 v2 v2 v4 v4 v2 v3 v5 v3 v5 dessin non planaire de G un dessin planaire de G v3 12 Major results (on planar graphs) in graph theory Thm (Steinitz, 1916) F´arytheorem (1947) • Every (simple) planar graph admits a straight line planar embedding (no edge crossings) v1 v1 v1 v2 v2 v4 v4 v2 v3 v5 v3 v5 dessin non planaire de G un dessin planaire de G v3 12 Major results (on planar graphs) in graph theory Thm (Steinitz, 1916) 3-connected planar graphs are the 1- skeletons of convex polyhedra Thm (Whitney, 1933) 3-connected planar graphs admit a unique planar embedding (up to homeomorphism and inversion of the sphere). 12 Major results (on planar graphs) in graph theory Thm (Steinitz, 1916) 3-connected planar graphs are the 1- skeletons of convex polyhedra Thm (Whitney, 1933) 3-connected planar graphs admit a unique Def G is 3-connected if planar embedding (up to homeomorphism and is connected and inversion of the sphere). the removal of one or two vertices does not disconnect G at least 3 vertices are required to disconnect the graph 12 Major results (on planar graphs) in graph theory Thm (Koebe-Andreev-Thurston) Every planar graph with n vertices is isomorphic to Schnyder woods (via the intersection graph of n disks in the plane. dimension of partial or- ders) • dim(G) ≤ 3 Thm (Colin de Verdi`ere,1990) Theorem (Lovasz Schrijver ’99) Given a 3-connected planar graph G, the eigenvectors ξ ; ξ ; ξ of a CdV Colin de Verdiere invariant (multiplicity of λ eigen- 2 3 4 2 matrix defines a convex polyhedron containing the origin.. value of a generalized laplacian) N • µ(G) ≤ 3 N S2 v M v 4 −1 ::: ::: 0 deg(vi) −1 5 ::: LG[i; k] = ::: ::: f −AG[i; j] ::: ::: 0 ::: 3 12 Major results (on planar graphs) in graph theory Thm (Tutte barycentric method, 1963) Every 3-connected planar graph G admits a barycentric representation ρ in R2. X P ρ(vi) = wijρ(vj) ( j wij = 1 and wij > 0 j2N(i) 2 ρ :(VG) −! R is barycentric iff for each inner node vi, ρ(vi) is the barycenter of Get a straight line drawing the images of its neighbors solving a system a linear equations N(v4) = fv1; v2; v3; v5g N(v5) = fv2; v3; v4g 3 −1 −1 −1 0 −1 4 −1 −1 −1 M · x = ax v1 v1 L = −1 −1 4 −1 −1 v2 f M · y = ay v4 −1 −1 −1 4 −1 0 −1 −1 −1 3 v2 v3 v5 laplacian matrix v3 12 Major results (on planar graphs) in graph theory Theorem (Schnyder ’89) A graph G is planar if and only if the di- mension of its incidence poset is at most 3 Theorem (Schnyder, Soda ’90) For a triangulation T having n vertices, we can draw it on a grid of size (2n − 5) × (2n − 5), by setting v0 = (2n − 5; 0), v1 = (0; 0) and v2 = (0; 2n − 5). v2 (a) (b) v5 v4 v3 v0 v1 v3 (1; 2; 4) v4 (2; 4; 1) v0 v1 v5 (4; 1; 2) 12 Part II What is a surface mesh? (a short digression on embedded graphs, simplicial complexes and topological and combinatorial maps) 13 What is a (surface) mesh? surface mesh: set of vertices, edges and faces (polygons) defining a polyhedral surface in embedded in 3D (discrete approximation of a shape) Combinatorial structure + geometric embedding ”Connectivity”: the underlying map vertex coordinates incidence relations between triangles, vertices and edges 14 Planar and surface meshes: definition planar triangulation planar triangulation planar map embedded in R3 spherical drawing straight line drawing of a dodecahedron triangle mesh spherical parameterizations of a triangle mesh (Gotsman, Gu Sheffer, 2003) toroidal map (Eric Colin de Verdiere)` 15 Surface meshes as simplicial complexes abstract simplicial complex K (set of simplices) V = fv0; v1; : : : ; vn−1g E = ffi; jg; fk; lg;:::g F = ffi; j; kg; fj; i; lg;:::g inclusion property: ρ 2 K and σ ⊂ ρ −! σ 2 K not valid simplicial complex intersection property: given two simplices σ1; σ2 of K, the intersection σ1 [ σ2 is a face of both 16 Surface meshes as (topological) maps (geomettric realizations of maps) two different embeddings of the same graph A graph G = (V; E) is a pair of: • a set of vertices V = (v1; : : : ; vn) • a collection of E = (e1; : : : ; em) elements of the cartesian product V × V = f(u; v) j u 2 V; v 2 V g (edges). cellular embeddings of a graph defining the same (planar) map a planar drawing is a cellular embedding of G into R2, satisfying: (i) graph vertices are represented as points ; (ii) edges are represented as no croissing curves ; (iii) faces are simply connected. (topologica) map: cellular embedding up to homeomorphism (equivalence class) two cellular embeddings defining the same planar map 17 Surface meshes as combinatorial maps (geomettric realizations of maps) 17 22 23 18 3 permutations on the set H of the 2n half-edges 20 2 10 19 11 6 1 3 5 8 21 24 (i) α involution without fixed point; 14 4 12 15 (ii) ασφ = Id; 7 16 13 9 (iii) the group generated by σ, α et φ transitively on H. φ = (1; 2; 3; 4)(17; 23; 18; 22)(5; 10; 8; 12)(21; 19; 24; 15) ::: α = (2; 18)(4; 7)(12; 13)(9; 15)(14; 16)(10; 23) ::: 18 Surface meshes as combinatorial maps (geomettric realizations of maps) 17 22 23 18 3 permutations on the set H of the 2n half-edges 20 2 10 19 11 6 1 3 5 8 21 24 (i) α involution without fixed point; 14 4 12 15 (ii) ασφ = Id; 7 16 13 9 (iii) the group generated by σ, α et φ transitively on H.
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