Realizing Planar Graphs as Convex Polytopes G unter Rote Freie - PowerPoint PPT Presentation

Realizing Planar Graphs as Convex Polytopes G¨ unter Rote Freie Universit¨ at Berlin G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

General Problem Statement GIVEN: CONSTRUCT: a combinatorial type of a geometric realization of 3-dimensional polytope the polytope (a 3-connected planar graph) [ + additional data ] [ with additional properties ] G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

General Problem Statement GIVEN: CONSTRUCT: a combinatorial type of a geometric realization of 3-dimensional polytope the polytope (a 3-connected planar graph) [ + additional data ] [ with additional properties ] e.g.: small integer vertex coordinates G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Polytopes with Small Vertex Coordinates Every polytope with n vertices can be realized with integer coordinates less than 148 n . [ Rib´ o, Rote, Schulz 2011, Buchin & Schulz 2010 ] Lower bounds: Ω( n 1 . 5 ) Better bounds for special cases: O ( n 18 ) for stacked polytopes [ Demaine & Schulz 2011 ] G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Schlegel Diagrams O project from a center O close enough to a face G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Schlegel Diagrams O a Schlegel diagram: project from a center O a planar graph with close enough to a face convex faces G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

3-Connectivity Assume a, b separate the graph G . Choose a third vertex v . Take a plane π through a, b, v . π v a b G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

3-Connectivity Assume a, b separate the graph G . Choose a third vertex v . Take a plane π through a, b, v . v max Every vertex has a monotone path to v max or v min . π v has both paths. v a b = ⇒ G − { a, b } is connected. d -connected in d dimensions [ Balinski 1961 ] v min [ this proof: Gr¨ unbaum ] G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

The Theorem of Steinitz (1916) The graphs of convex three-dimensional polytopes are exactly the planar , 3-connected graphs. We have seen “ = ⇒ ”. Whitney’s Theorem: 3-connected planar graphs have a unique face structure. ( = ⇒ they have a combinatorially unique plane drawing up to reflection and the choice of the outer face.) = ⇒ The combinatorial structure of a 3-polytope is given by its graph. G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Constructive Approaches 1. INDUCTIVE Start with the simplest polytope and make local modifications. [ Steinitz ] [ Das & Goodrich 1995 ] 2. DIRECT Obtain the polytope as the result of • a system of equations [ Tutte ] • an optimization problem � • an iterative procedure [ Koebe–Andreyev–Thurston ] • (and existential argument) G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

The Realization Space assume: origin in the interior of P . n vertices, m edges, f faces ( a j , b j , c j ) P a j x + b j y + c j z ≤ 1 ( x i , y i , z i ) G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

The Realization Space assume: origin in the interior of P . n vertices, m edges, f faces ( a j , b j , c j )   x 1 y 1 z 1 x 2 y 2 z 2     P . . .   a j x + b j y + c j z ≤ 1   x n y n z n     a 1 b 1 c 1     a 2 b 2 c 2   ( x i , y i , z i )   . . .   a f b f c f � = 1 , if face j contains vertex i ( a j , b j , c j ) · ( x i , y i , z i ) < 1 , otherwise G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

The Realization Space x 1 y 1 z 1 n vertices, m edges, f faces   x 2 y 2 z 2   . . .   R 0 = { ∈ R ( n + f ) × 3 : x n y n z n     a 1 b 1 c 1   a 2 b 2 c 2     . . . a f b f c f � = 1 , if face j contains vertex i ( a j , b j , c j ) · ( x i , y i , z i ) < 1 , otherwise 3 n + 3 f variables, 2 m equations THEOREM: dim R 0 = 3 n + 3 f − 2 m = m + 6 . R 0 is contractible. In 4 and higher dimensions, realization spaces can be arbitrarily complicated. [ Mn¨ ev 1988, Richter-Gebert 1996 ] G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

The Realization Space x 1 y 1 z 1 n vertices, m edges, f faces   x 2 y 2 z 2   . . .   R 0 = { ∈ R ( n + f ) × 3 : x n y n z n     a 1 b 1 c 1   a 2 b 2 c 2     . . . a f b f c f � = 1 , if face j contains vertex i ( a j , b j , c j ) · ( x i , y i , z i ) < 1 , otherwise • triangulated (simplicial) polytopes vertices can be perturbed. ( a j , b j , c j ) variables are redundant. G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

The Realization Space x 1 y 1 z 1 n vertices, m edges, f faces   x 2 y 2 z 2   . . .   R 0 = { ∈ R ( n + f ) × 3 : x n y n z n     a 1 b 1 c 1   a 2 b 2 c 2     . . . a f b f c f � = 1 , if face j contains vertex i ( a j , b j , c j ) · ( x i , y i , z i ) < 1 , otherwise • simple polytopes (with 3-regular graphs) faces can be perturbed. ( x i , y i , z i ) variables are redundant. G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

The Realization Space x 1 y 1 z 1 n vertices, m edges, f faces   x 2 y 2 z 2   . . .   R 0 = { ∈ R ( n + f ) × 3 : x n y n z n     a 1 b 1 c 1   a 2 b 2 c 2     . . . a f b f c f � = 1 , if face j contains vertex i ( a j , b j , c j ) · ( x i , y i , z i ) < 1 , otherwise Polarity: interpret ( a j , b j , c j ) as vertices and ( x i , y i , z i ) as half-spaces. → the polar polytope: VERTICES ↔ FACES exchange roles. → the (planar) dual graph G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Inductive Constructions of Polytopes G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Inductive Constructions of Polytopes G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Inductive Constructions of Polytopes an additional (triangular) face + apply polarity when necessary [ Steinitz 1916 ] Everything can be done with rational coordinates. → integer coordinates of size 2 exp( n ) COMBINATORIAL + GEOMETRIC arguments G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Inductive Constructions of Polytopes Das & Goodrich [1997]: 2 poly( n ) for triangulated polytopes perform this operation on n/ 24 independent vertices in parallel → O (log n ) rounds Each round multiplies the number of bits by a constant factor. G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Inductive Constructions of Polytopes Das & Goodrich [1997]: 2 poly( n ) for triangulated polytopes perform this operation on n/ 24 independent vertices in parallel → O (log n ) rounds Each round multiplies the number of bits by a constant factor. G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

Direct Constructions of Polytopes A) construct the Schlegel B) Lift to three dimensions. diagram in the plane. G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

When is a Drawing a Schlegel Diagram? strictly convex faces! G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

When is a Drawing a Schlegel Diagram? strictly convex faces! 1 2 3 G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011

When is a Drawing a Schlegel Diagram? strictly convex faces! 1 2 3 G¨ unter Rote, Freie Universit¨ at Berlin Realizing Planar Graphs as Convex Polytopes Graph Drawing, Eindhoven, 21.–23. 9. 2011