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

Marta Panizzut

November 24, 2020

1 / 20 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.

“only if”: radial projection to the sphere from an interior point or Schlegel diagram + Balinski’s theorem.

2 / 20 Theorem 9 Let G be a graph. (i) [Menger’s Theorem] If G is 3-connected, then between any pair of vertices there are three paths G1, G2 and G3 that are disjoint except for the endpoints. (ii) If there are three vertices u,v,w that are connected by three different Y -graphs Y1, Y2 and Y3 that are disjoint except for u, v and w, then G is not planar. (iii) [Whitney’s Theorem] If G is planar and 3-connected then the set cells(G ,D) is independent on the particular choice of a drawing D of G. (iv) [Euler’s Theorem] If G is planar and 2-connected and D is a drawing of G, then the numbers of cells c cells(G ,D) , = | | vertices v V , and edges e E are related by = | | = | | v c e 2. + = +

3 / 20 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]

4 / 20 Notation G is planar and 3-connected with vertices labeled by V {1,...,n} so = that the last k 1 vertices c0 (k 1,...,n) are a cell of G. + = + We realize c0 as boundary cell.

5 / 20 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 − = ∈

6 / 20 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.

7 / 20 Proof of existence and uniqueness

We know the positions of the peripheral vertices pv (xv ,yv ), Ï = v {k 1,...,n}. We need the positions of the interior vertices ∈ + pv (xv ,yv ), v {1,...,k}. We assume pn (0,0). = ∈ = E(x ,...,x ,y ,...,y ) 1 P ¡(x x )2 (y y )2¢ 1 k 1 k 2 {v,w} E 0 ωv,w v w v w Ï = ∈ − + − is quadratic and non-negative everywhere. For sufficiently large α 0, z α implies E(z) E(0). Ï > | | > > E is strictly convex and takes its unique minimum on Ï {z z α}. || | < Critical point Ï ∂E X ∂E X ωv,w (xv xw ) 0 ωv,w (yv yw ) 0 ∂xi = {v,w} E − = ∂yi = {v,w} E − = ∈ 0 ∈ 0

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8 / 20 Relative interior 2 The relative interior of a collection of points P : (p1,...,pn) R is = ⊆ defined by

n n n X X o relint(P) λv pv λv 1 and λv 0 for all v 1,...,n . = v 1 | v 1 = > = = = Lemma 12 2 Let p relint(p1,...,pn) and let φ be a linear functional on R . ∈ (i) If there exists v {1,...,n} with φ(p) φ(pv ) then there exists ∈ < also a w {1,...,n} with φ(p) φ(pw ) ∈ 2> (ii) If {p1,...,pn} affinely spans R then

relint(p1,...,pn) int(conv(p1,...,pn)). = Lemma 13 2 The set of configurations (p0,p1,...,pn) of points in R for which 2 p1,...,pn affinely span R , and p0 relint(p1,...,pn) is an open 2(n 1) ∈ subset of R + . 9 / 20 Lemma 14 Let P : {p1,...,pn} be an equilibrium representation of the vertices = of G. Then for every interior vertex p we have p relint(N(p)). ∈

Proof. Let pv be an interior vertex. The equilibrium condition is X ωv,w (pv pw ) 0. {v,w} E − = ∈ This rewrites

1 ³ X ´ pv P ωv,w pw . = {v,w} E ωv,w {v,w} E ∈ ∈

So pv is convex combination of its neighbors with strictly positive coefficients.

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10 / 20 Good representations

Definition 15 Let G be a 3-connected, planar graph on n vertices such that the ordered vertices k 1,...,n are a cell in G. A point + configuration P (p1,...,pn) is a good representation for G, if the = following properties are satisfied

(i) pk 1,...,pn realize a convex (n k)-gon, + − (ii) For v 1,...,k we have pv relint(N(pv )). = ∈ Remark: Equilibrium representations are good representations.

11 / 20 Theorem 16 Let P be a good representation of a 3-connected, planar graph G (V ,E), then P is a planar embedding of G in which all interior = cells are realized as non-overlapping convex polygons.

12 / 20 Claim 1 Let P be a good representation of a 3-connected, planar graph G. Then P has no degenerate vertices. A vertex pv is degenerate if its set of neighbors N(pv ) lie on a line.

13 / 20 Claim 2 For a given 3-connected planar graph G the set of all good represen- 2n 2n tations P : (p1,...,pn) R is an open subsets of R . = ∈

14 / 20 Angles

Let G 0 be a 3-connected, planar graph, for which at most one cell is not a triangle. G 0 has n vertices and k 1,...,n forms the possibly + non-triangular cell. We consider a planar embedding and around each vertex v we take 1 N(v) the counterclockwise cyclic order (v ,...,v | |) of its neighbors.

Let P be a very good representation of G 0 (no three points of a i triangle are collinear). We set α (pv ) (pv ;p i ,p i 1 ), the angle = ∠ v v + between p i and p i 1 seen from p . We define v v + v

N(v) | X | i Ω(pv ) α (pv ) . = i 1 | | =

15 / 20 Claim 3 Let P be a very good representation of G 0. (i) If pv is a peripheral point then Ω(pv ) 2 βv , where βv is the ≥ | | angle between the two neighbors of pv in the peripheral cell. (ii) If pv is an interior point, then Ω(pv ) 2π. ≥ Claim 4 There are exactly n k 2 interior cells. + − Claim 5 Let P be a very good representation of G 0. Pn Then v 1 Ω(pv ) (2n 4)π = = − Claim 6 Let P be a very good representation of G 0 then for every interior point pv , we have Ω(pv ) 2π. = For every peripheral point pv we have Ω(pv ) 2 βv . = | |

16 / 20 Claim 7 For three non-collinear points a,b,c the angles ∠(a;b,c), ∠(b;c,a) and ∠(c;a,b) are either all positive or all negative.

Claim 8 Let P be a very good representation of G 0. Except for the angles i between peripheral points all α (pv ) are positive.

We call these interior angles.

17 / 20 Proof of Theorem

Let G (V ,E) be a planar and 3-connected graph together with a = peripheral cell and let P be a good representation of G.

We add interior edges to G to obtain a graph G 0 (V ,E 0) that Ï = it is still planar but for which all interior edges are .

We represent triangles of G 0 by oriented 3-cycles Ï 1 2 3 Ti (v ,v ,v ). We choose orientation consistently. = i i i

18 / 20 Proof of Theorem

In the representation P the vertices of each triangle of G are Ï 0 affinely independent. So we have a very good representation and by Claim 8 all Ï interior angles are strictly positive. We consider the oriented area vol(T ). Area positive if Ï (pv1 ,pv2 ,pv3 ) come in counterclockwise order. Again by Claim 8 all the areas are positive. Pt vol(c0) vol(Ti ). This implies that two triangles do not Ï = i 1 overlap. = If we only consider the edges of G we get a drawing with cells Ï realized as non-overlapping polygons. Each internal vertex lies in the relative interior of its neighbors Ï the cells must be convex.

■ Details of the proofs can be found in [RG96, §12].

19 / 20 References

Jürgen Richter-Gebert, Realization spaces of polytopes, Lecture Notes in Mathematics, vol. 1643, Springer-Verlag, Berlin, 1996.

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