INF562, Lecture 3: Geometric and combinatorial properties of planar graphs mardi 22 janvier 2013 Luca Castelli Aleardi
Intro Graph drawing: motivations and applications
Graph drawing and data visualization Global transportation system
Graph drawing and data visualization Roads, railways, ...
Graph drawing and data visualization Social network graph
Planar graphs french roads network Design of integrated circuits (VLSI)
Planar graphs french roads network Design of integrated circuits (VLSI) www.2m40.com 9 accidents en 2012 (last one, on 28th sptember)
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
Mesh parameterization in geometry processing
Mesh parameterization in geometry processing
Mesh parameterization in geometry processing
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Graph drawing: motivation Challenge: what kind of graph does A G represent? 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 A G = 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0
Part I Major results in graph theory
Major results (on planar graphs) in graph theory
Major results (on planar graphs) in graph theory Kuratowski theorem (1930) (cfr Wagner’s theorem, 1937) • G contains neither K 5 nor K 3 , 3 as minors
Major results (on planar graphs) in graph theory Kuratowski theorem (1930) (cfr Wagner’s theorem, 1937) • G contains neither K 5 nor K 3 , 3 as minors F´ ary theorem (1947) • Every (simple) planar graph admits a straight line planar embedding (no edge crossings) v 1 v 1 v 1 v 2 v 2 v 4 v 4 v 2 v 5 v 5 v 3 v 3 dessin non planaire de G un dessin planaire de G v 3
Major results (on planar graphs) in graph theory Thm (Steinitz, 1916) F´ ary theorem (1947) • Every (simple) planar graph admits a straight line planar embedding (no edge crossings) v 1 v 1 v 1 v 2 v 2 v 4 v 4 v 2 v 5 v 5 v 3 v 3 dessin non planaire de G un dessin planaire de G v 3
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).
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 inversion of the sphere). is connected and the removal of one or two vertices does not disconnect G at least 3 vertices are required to disconnect the graph
Major results (on planar graphs) in graph theory Thm (Koebe-Andreev-Thurston) Every planar graph with n vertices is isomorphic to the intersection graph of n disks in the plane. Thm (Colin de Verdi` ere, 1990) Theorem (Lovasz Schrijver ’99) Given a 3 -connected planar graph G , the eigenvectors ξ 2 , ξ 3 , ξ 4 of a CdV Colin de Verdiere invariant (multiplicity of λ 2 eigen- matrix defines a convex polyhedron containing the origin. . value of a generalized laplacian) • µ ( G ) ≤ 3 λ 1 = − 4 , λ 2 = λ 3 = λ 4 = 0 v 2 M ξ x ξ y ξ z − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 v 0 v 3 1 − 1 − 1 − 1 − 1 0 0 v 1 0 0 1 v 2 − 1 − 1 − 1 − 1 1 v 3 0 0 v 0 v 1
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 R 2 . � ρ ( v i ) = w ij ρ ( v j ) ( � j w ij = 1 and w ij > 0 j ∈ N ( i ) → R 2 is barycentric iff for each ρ : ( V G ) − inner node v i , ρ ( v i ) is the barycenter of Get a straight line drawing the images of its neighbors solving a system a linear equations N ( v 4 ) = { v 1 , v 2 , v 3 , v 5 } N ( v 5 ) = { v 2 , v 3 , v 4 } 3 − 1 − 1 − 1 0 { M · x = a x − 1 4 − 1 − 1 − 1 v 1 v 1 L = M · y = a y − 1 − 1 4 − 1 − 1 v 2 v 4 − 1 − 1 − 1 4 − 1 0 − 1 − 1 − 1 3 v 2 v 5 laplacian matrix v 3 v 3
Major results (on planar graphs) in graph theory (Schnyder ’89) Theorem 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 (2 n − 5) × (2 n − 5) , by setting v 0 = (2 n − 5 , 0) , v 1 = (0 , 0) and v 2 = (0 , 2 n − 5) . v 2 (b) (a) v 5 v 4 v 3 v 1 v 0 v 3 (1 , 2 , 4) v 4 (2 , 4 , 1) v 1 v 0 v 5 (4 , 1 , 2)
Part II What is a surface mesh? (a short digression on embedded graphs, simplicial complexes and topological and combinatorial maps)
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
Planar and surface meshes: definition planar triangulation planar triangulation planar map embedded in R 3 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 Verdi` ere)
Surface meshes as simplicial complexes abstract simplicial complex K (set of simplices) V = { v 0 , v 1 , . . . , v n − 1 } E = {{ i, j } , { k, l } , . . . } F = {{ i, j, k } , { j, i, l } , . . . } inclusion property : not valid simplicial complex ρ ∈ K and σ ⊂ ρ − → σ ∈ K intersection property : given two simplices σ 1 , σ 2 of K , the intersection σ 1 ∪ σ 2 is a face of both
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 = ( v 1 , . . . , v n ) • a collection of E = ( e 1 , . . . , e m ) elements of the cartesian product V × V = { ( u, v ) | u ∈ V, v ∈ V } ( edges ). cellular embeddings of a graph defining the same (planar) map un dessin planaire est un plongement cellulaire de G dans R 2 , qui satisfait les conditions suivantes: (i) les sommets du graphe sont repr´ sent´ es par des points ; (ii) les aretes sont repr´ sent´ ees par des arcs de courbes ne se coupant qu’aux sommets ; (iii) les faces sont simplement connexes. (topologica) map : cellular embedding up to homeomorphism (equivalence class) two cellular embeddings defining the same planar map
Surface meshes as combinatorial maps (geomettric realizations of maps) 17 22 23 3 permutations on the set H of the 2 n half-edges 18 20 6 2 3 108 19 (i) α involution without fixed point; 11 1 5 21 24 14 4 12 15 (ii) ασφ = Id ; 7 13 16 (iii) the goup generated by σ , α et φ transitively on H . 9 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) . . .
Surface meshes as combinatorial maps (geomettric realizations of maps) 17 22 23 3 permutations on the set H of the 2 n half-edges 18 20 6 2 3 108 19 (i) α involution without fixed point; 11 1 5 21 24 14 4 12 15 (ii) ασφ = Id ; 7 13 16 (iii) the goup generated by σ , α et φ transitively on H . 9 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) . . .
Mesh representations: classification Manifold meshes non manifold or non orientable meshes triangle meshes no boundary genus 1 mesh with boundaries Manifold mesh: definition Every edge is shared by at most 2 faces For every vertex v , the incident faces form an open or closed fan v quad meshes polygonal meshes
Part III Euler formula and its consequences
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