# Proof of the two paths theorem

{pause up}
## Generating (hyper)webs inductively

{pause .theorem}
The only web with a frame of length $3$ is the triangle.

{paues .proof}
The frame is a triangle, and hence an inner face on top of being the outer face.

{pause .block title="Problem" #problem-1}
Can webs be inductively generated from the triangle ?

{pause #parallel}
Parallel composition:

> {pause carousel #operation-parallele}
> ----
> > ![](dessins/operation-parallel-G-H.svg)
> ---
> > ![](dessins/operation-parallel.svg)
> ---
> > ![](dessins/operation-parallel-edges.svg)
> ---
> > ![](dessins/operation-parallel.svg)

{change-page="operation-parallele"}

{pause}
- $G$ and $H$ have the same number of squares vertices, {pause}
- square vertices must be ordered somewhat.

{pause .definition title="Web compositions" up="parallel" #web-composition-def}
The **(hyper)web composition** of two (hyper)webs $W_1$ and $W_2$ is their parallel composition along induced subpaths of their respective frames.

{pause}
{pause carousel #web-composition}
----
> ![](dessins/web-composition-1.svg)
---
> ![](dessins/web-composition-2.svg)
---
> ![](dessins/hyperweb-composition-1.svg)
---
> ![](dessins/hyperweb-composition-2.svg)

{change-page="web-composition"}

{change-page="web-composition"}

{change-page="web-composition"}

{change-page="operation-parallele"}

{change-page="operation-parallele"}

{pause .theorem up="web-composition-def" #web-composition-theo}
> 1. (Hyper)webs are stable under (hyper)web compositions.
>
> 2. Any (hyper)web with a frame of length at least four is the (hyper)web composition of two (hyper)webs.

{pause .proof up="web-composition"}
> {pause}
> 1. Compose planar drawings whose only common parts are the identified induced subpaths of the frames. {pause up="web-composition-theo"}
> 2.
>
> {pause carousel #web-comp-proof}
> ----
> > ![](dessins/web-comp-proof-1.svg)
> ---
> > ![](dessins/web-comp-proof-2.svg)
> ---
> > ![](dessins/web-comp-proof-3.svg)
> ---
> > ![](dessins/web-comp-proof-4.svg)
> ---
> > ![](dessins/web-comp-proof-5.svg)
> ---
> > ![](dessins/web-comp-proof-6.svg)

{change-page="web-comp-proof"}

{change-page="web-comp-proof"}

{change-page="web-comp-proof"}

{change-page="web-comp-proof"}

{change-page="web-comp-proof"}

<br>

{pause down}
**The set of graph obtained from the triangle (respectively a $3$-edge) by successive (hyper)web compositions is precisely the set of all (hyper)webs.**

{pause up}
## Proof of the two paths theorem

{.theorem #theoreme-deux-chemins-deux title="Two paths theorem"}
> Let $G$ be a graph and $C$ a cycle on $G$'s vertices. Either $G$ contains a $C$-crossing, or $G$ has the shape of an hyperweb with frame $C$.
>
> Furthermore, with $C$ the frame of an hyperweb $H$,
> - there is no $C$-crossing in $H$,
> - if $u$ and $v$ are two non-adjactent vertices of $H$, then $H+uv$ contains a $C$-crossing.

{pause .proof #preuve-theoreme-deux-chemin}
> We only sketch: no $C$-crossing $\Rightarrow$ shape of an hyperweb with frame $C$.
>
> {pause carousel #preuve-deux-chemins}
> ----
> > ![](dessins/preuve-deux-chemins-graphe.svg)
> ---
> > ![](dessins/preuve-deux-chemins-P.svg)
> ---
> > ![](dessins/preuve-deux-chemins-cross.svg)
> ---
> > ![](dessins/preuve-deux-chemins-barre.svg)
> ---
> > ![](dessins/preuve-deux-chemins-separation.svg)
> ---
> > ![](dessins/preuve-deux-chemins-mesure.svg)
> ---
> > ![](dessins/preuve-deux-chemins-final.svg)
> ---
> > ![](dessins/preuve-deux-chemins-composition.svg)
>
> {change-page="preuve-deux-chemins"}
>
> {change-page="preuve-deux-chemins"}
>
> {change-page="preuve-deux-chemins"}
>
> {change-page="preuve-deux-chemins"}
>
> {change-page="preuve-deux-chemins"}
>
> {change-page="preuve-deux-chemins"}
>
> {change-page="preuve-deux-chemins" up="theoreme-deux-chemins-deux"}
>