Add Even Level Graph Generators paper + extend Level Switching reachability

- New paper papers/even_level_graph_generators/: defines Even Level
  Graph (every level cycle even), derived level graphs, intertwining
  trees, and the disjunction conjecture (every maximal planar graph is
  a derived level graph or intertwining tree). Empirically tested
  through n=11: every iso class is at least an intertwining tree, so
  the disjunction holds trivially in this range. The intertwining tree
  disjunct fails at the Tutte graph dual (n=25), so the disjunction
  becomes non-trivial past some unknown threshold.

- Level Switching paper: adds Section 4 (Reachability via edge
  switches) with the two-step argument (Sleator-Tarjan-Thurston for
  Case 1; face-merges for Case 2) and Theorem 4.1 (O(n) edge switches
  suffice to reach all-depth-0).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/level_switching/paper.tex

@@ -561,4 +561,107 @@ reached from the current maximum-depth face by a preprocessing path of
 length bounded by the dual-tree diameter of $L_k$?
 \end{question}
 
+\section{Reachability via edge switches}
+\label{sec:reachability}
+
+We now sketch a positive termination argument that bypasses the local
+question of balanced surface switches entirely: instead of insisting
+that each switch strictly improve facial depth, we show that any
+$L_k$ can be transformed by edge switches into a configuration in which
+every face has depth $0$. Throughout this section we adopt the
+\emph{stable-labelling convention}: the level $\ell_G(v)$ of each
+vertex is fixed at the start of the procedure (by BFS from $S$ in the
+initial triangulation $G$) and reused thereafter, even after edge
+switches modify the triangulation. In particular, the level-$k$
+subgraph $L_k$ of the current triangulation always means ``the
+subgraph induced on the vertices labelled $k$ at the start''.
+
+\subsection*{Two cases on the layer below $k$}
+
+We split on whether any $L_k$-face has a higher-level vertex in its
+interior in the planar embedding inherited from $\Pi_G$.
+
+\textbf{Case 1: every inner face of $L_k$ is a triangle and contains
+no vertex of level $\geq k+1$ in its interior.}
+
+Under this hypothesis, for every chord $uv \in L_k$ the two $G$-triangles
+at $uv$ have their third vertices in $L_k$ (since the interior of the
+two $L_k$-faces adjacent to $uv$ in $\Pi_G$ contains no other vertex of
+$G$). The edge switch at $uv$ is therefore always in Case~(ii) of
+Proposition~\ref{prop:balanced-descent}, and acts as a flip of the
+chord $uv$ in $L_k$ regarded purely as a maximal outerplanar graph.
+
+Maximal outerplanar graphs on $n$ labelled vertices (arranged on a
+common outer cycle) are exactly triangulations of a convex $n$-gon.
+The set of such triangulations, with chord flips as edges, is the
+1-skeleton of the associahedron and is connected; in fact any two
+triangulations are joined by $O(n)$ chord flips~\cite{sleator-tarjan-thurston}.
+A \emph{fan triangulation} -- the triangulation obtained by adding
+chords from a single apex vertex $v_0$ to every other vertex -- has
+every inner triangle bounded by an outer-cycle edge (namely the side
+opposite $v_0$ in that triangle), so every face of a fan triangulation
+lies in $\mathcal{B}$ and has depth $0$.
+
+Combining: in Case~1, $L_k$ can be transformed into a fan triangulation
+by $O(n)$ edge switches, after which every face has depth $0$.
+
+\textbf{Case 2: some $L_k$-face $F$ has a vertex of level $\geq k+1$
+in its interior.}
+
+Pick any edge $uv$ of $\partial F$. The $G$-triangle at $uv$ on the
+$F$-side has its third vertex $w$ inside $F$, so $w$ is a vertex of
+level $\geq k+1$ and in particular $w \notin L_k$. The edge switch
+at $uv$ is therefore in Case~(i) of
+Proposition~\ref{prop:balanced-descent}: the edge $uv$ is removed from
+$L_k$, no new edge is added to $L_k$, and the face $F$ merges with the
+$L_k$-face on the opposite side of $uv$ into a single larger face. The
+number of inner faces of $L_k$ strictly decreases by $1$.
+
+\subsection*{Combining}
+
+\begin{theorem}
+\label{thm:reachability}
+Under the stable-labelling convention, every $L_k$ can be transformed
+by edge switches into a configuration in which every inner face of
+$L_k$ has facial depth $0$, in $O(n)$ edge switches.
+\end{theorem}
+
+\begin{proof}[Proof sketch]
+Apply Case~2 repeatedly while $L_k$ has any inner face with a
+higher-level vertex in its interior. Each application reduces the
+number of inner faces of $L_k$ by $1$, so after at most $|L_k| - 2 \leq
+n - 2$ such steps we reach one of:
+\begin{itemize}
+\item A configuration satisfying the hypothesis of Case~1, in which
+case Case~1 finishes the job in $O(n)$ flips by reaching a fan
+triangulation.
+\item A configuration in which $L_k$ has only one inner face -- i.e.,
+$L_k$ consists of only its outer cycle, with no chords. The unique
+inner face is bounded by all $n$ outer-cycle edges, so it lies in
+$\mathcal{B}$ and has depth $0$.
+\end{itemize}
+Both outcomes leave every face at depth $0$. The total step count is
+at most $(n - 2) + O(n) = O(n)$.
+\end{proof}
+
+\begin{remark}
+Theorem~\ref{thm:reachability} settles the existence question
+Question~\ref{q:preprocessing-terminates} affirmatively in the
+following sense: \emph{some} sequence of edge switches drives every
+face to depth $0$ in $O(n)$ steps. It does not, however, identify the
+sequence by a local rule (the leaf-distance algorithm of
+Section~\ref{sec:reachability}'s preceding discussion), and in
+particular the question of which \emph{rule} produces such a sequence
+without backtracking remains open.
+\end{remark}
+
+\begin{thebibliography}{9}
+
+\bibitem{sleator-tarjan-thurston}
+D.~D. Sleator, R.~E. Tarjan, W.~P. Thurston.
+\emph{Rotation distance, triangulations, and hyperbolic geometry}.
+Journal of the American Mathematical Society, 1988.
+
+\end{thebibliography}
+
 \end{document}