Steinitz’s Theorem, II Realization Spaces of Winter Semester 2020/2021

Marta Panizzut

December 1, 2020

1 / 23 Steinitz’s Theorem Theorem 8 [Steinitz ’22] A finite graph is the edge-graph of a 3- if and only if it is simple, planar and 3-connected.

Idea of the proof: Draw a 3-connected such that its edges are Ï represented by line segments, and the cells are convex .

If the boundary of the drawing of G is a , then the Ï figure is a Schlegel diagram of a 3-polytope with edge-graph G.

Picture from [RG96, §12]

2 / 23 Notation G is planar and 3-connected with vertices labeled by V {1,...,n}. = The last k 1 vertices c0 (k 1,...,n) are a cell of G. + = + We realize c0 as boundary cell.

3 / 23 Equilibrium

Let G be a graph. To each edge {v,w} E we assign a weight ∈ ωv w R. We impose the symmetry condition ωv w ωw v . , ∈ , = , Definition 10 Let G (V ,E) be a graph and ω : E R be an assignment of weights. = → Furthermore, let p : V R2 be an assignment of positions in R2 for → the vertices of G. A v V is in equilibrium if ∈ X ωv,w (pv pw ) 0. {v,w} E − = ∈

4 / 23 Tutte’s Theorem

Theorem 11 [Tutte ’62] Let G ({1,...,n}, E) be a 3-connected, planar graph that has a cell = (k 1,...,n) for some k n. Let pk 1,...,pn be the ordered vertices + < + of a convex (n k)-gon. Let ω : E 0 R be an assignment of positive − → weights to the internal edges. 2 There are unique positions p1,...,pk R for the interior Ï ∈ vertices such that all interior vertices are in equilibrium. All cells c ,c ,... of G are then realized as non-overlapping Ï 1 2 convex polygons.

5 / 23 Setting

G conv(pk 1,...,pn). = +

We think the points embedded in R3 at the plane z 1. = So pi has coordinates (xi ,yi ,1). 6 / 23 Oriented patches

3 To each interior cell ci we associate a vector qi R by setting ∈ q1 (0,0,0) Ï = qL ωb t (pb pt ) qR if (b,t L,R) is an oriented patch. Ï = , × + |

Remark: For an oriented patch (b,t L,R), we have |

qL ωb t (pb pt ) qR qR ωt b(pt pb) qL = , × + ⇐⇒ = , × +

7 / 23 Lemma 17 The vectors qi are well defined. Proof.

j To a path Pi form a cell ci to a cell cj , we associate the vector

dPj qi qj . i = − i For the inverse path Pj we have dPj dPi . i = − j j j We need to prove that for two different paths Pi and Qi we have

dPj dQj (0,0,0). i − i =

8 / 23 It is enough to show this for a trip around an interior vertex p0. Let p1,...,p` be the cyclic sequence of its neighboring vertices:

P` Pl i 1 ω0,i (p0 pi ) i 1 ω0,i ((p0 pi ) (p0 p0)) = × = P`= × − × i 1 ω0,i (p0 (pi p0)) = = P` × − p0 i 1 ω0,i (pi p0) = × = − p0 (0,0,0) = × (0,0,0) =

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9 / 23 Lifting function We use the qi to construct a lifting function

f : G conv(pk 1,...,pn) R. = + → The function is defined as

f (x) x,qi if x ci . = 〈 〉 ∈

Lemma 18 The function is well defined. Proof. We show that the function agrees on pb and pt .

pb,qL pb,ωb t (pb pt ) qR ωb t pb,(pb pt ) pb,qR pb,qR . 〈 〉 = 〈 , × + 〉 = , 〈 × 〉+〈 〉 = 〈 〉

Then we need to show that the function agrees on every edge. It is enough to use that each point on an edge can be written as p λpb (1 λ)pt . = + − ■ 10 / 23 Local convexity

We have a unique height for each of the vertices. We need to show convexity. Lemma 19 For adjacent interior cells cL, and cR let ` be the line that supports the edge cL cR . Every point x that is on the same side of ` as cL ∩ satisfies x,qL x,qR . 〈 〉 > 〈 〉 Proof. Let (b,t L,R) be the oriented patch. We have |

x,qL x,qR x,ωb t (pb pt ) ωb t det(x,pb,pt ) 0 〈 〉 − 〈 〉 = 〈 , × 〉 = , · >

by orientation convention and ωb,t positive. ■

11 / 23 Global convexity

Lemma 20 At a point x int(ci ) the value x,qi is greater than all the values ∈ 〈 〉 x,qj with i j. 〈 〉 6= Proof. We take a line ` that connects x with an interior point y of cj , such that ` does not pass through the points in P. Moving from x to y we get a sequence of cells

ci cL ,cL ,...,cL cj . = 1 2 t = We have x,qi x,qL x,qj 〈 〉 > 〈 2 〉 > ··· > 〈 〉 ■

12 / 23 We have shown that the set

n 3 o n 3 o X x R x,qi 0 for i 1,...,m (x,y,z) R (x,y,1) G = ∈ |〈 〉 ≥ = ∩ ∈ | ∈ forms an unbounded polyhedral set.

Each of the equations x,qi 0 project down to the corresponding 〈 〉 = cell in the representation of G. The vertices of X are p0 (xi ,yi , pi ,qj ) where cj is a cell that i = 〈 〉 contains pi .

If the boundary cell is triangular, then the lifted peripheral vertices are coplanar.

13 / 23 Triangular cell Corollary 21 If a graph G is planar, 3-connected and contains a triangular cell, then it is the edge graph of a 3-polytope.

14 / 23 Triangular cell or vertex of degree 3

Lemma 22 A planar and 3-connected graph G contains either a triangular cell or a vertex of degree 3.

Proof. Let av be the average vertex degree and ac the average number of edges per cell. We have av v 2e and ac c 2e. Using Euler’s formula we get: = =

av v ac c 4e 4v 4c 8. + = = + −

Therefore one of av or ac is less than 4. Each vertex degree is at least 3 and each cell has at least 3 sides, there is a triangle or a vertex of degree 3.

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15 / 23 Polar graph

Lemma 23 Let P be a polytope and G be a 3-connected planar graph. G ∆ is a 3-connected planar graph. Ï G ∆∆ G. Ï = If G is the edge graph of P then G ∆ is the edge graph of P∆. Ï 16 / 23 Steinitz’s Theorem Theorem 8 [Steinitz ’22] A finite graph is the edge-graph of a 3-polytope if and only if is simple, planar and 3-connected.

17 / 23 Example

18 / 23 Example

19 / 23 Example

20 / 23 Different proofs

Classical approach: starting from a and iteratively Ï adding triangular faces and vertices of degree 3. ∆Y transformation in [Zie95, §4].

Koebe-Andreev-Thurston Circle Packing Theorem. Ï

21 / 23 Homework

Read combinatorial construction of pyramids, prisms, tents and Ï connected sum at [RG96, §3.1, §3.2].

22 / 23 References

Jürgen Richter-Gebert, Realization spaces of polytopes, Lecture Notes in Mathematics, vol. 1643, Springer-Verlag, Berlin, 1996. Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995.

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