plane_depth: add level/interlevel dual edge definitions

Extends Definition 1.2 (level edge) to also define "interlevel edge"
for the primal, and adds Definition 1.3 (level/interlevel dual edge)
classifying dual edges by whether they cross a level or interlevel
primal edge.

Useful downstream: in coloring_nested_tire_graphs, the partial tire
dual's edges can now be classified cleanly as level or interlevel
dual edges using the same vocabulary, instead of ad hoc "interior
annular edge" / "spoke edge" naming.

Paper stays at 4 pages.

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

papers/plane_depth/paper.aux

@@ -5,18 +5,18 @@
 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent }
 \newlabel{def:plane-depth}{{1.1}{1}}
 \newlabel{def:level-edge}{{1.2}{1}}
-\newlabel{def:triangle-types}{{1.3}{1}}
+\newlabel{def:level-dual-edge}{{1.3}{1}}
+\newlabel{def:triangle-types}{{1.4}{1}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Outerplanarity of depth levels}}{2}{}\protected@file@percent }
 \newlabel{lem:outerplanarity}{{2.1}{2}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Deep embedding}}{2}{}\protected@file@percent }
 \newlabel{def:deep-embedding}{{3.1}{2}}
-\newlabel{lem:up-down-faces}{{3.2}{2}}
-\bibcite{baker1994}{1}
+\newlabel{lem:up-down-faces}{{3.2}{3}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Quadrilateral decomposition}}{3}{}\protected@file@percent }
 \newlabel{lem:unique-level-edge}{{4.1}{3}}
 \newlabel{lem:shared-level-edge}{{4.2}{3}}
 \newlabel{def:quad-decomposition}{{4.3}{3}}
-\newlabel{def:quad-types}{{4.5}{3}}
+\bibcite{baker1994}{1}
 \bibcite{bauerfeld-pds-seq}{2}
 \bibcite{bauerfeld-nested-tires}{3}
 \newlabel{tocindent-1}{0pt}
@@ -24,5 +24,6 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
+\newlabel{def:quad-types}{{4.5}{4}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
 \gdef \@abspage@last{4}

papers/plane_depth/paper.log

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papers/plane_depth/paper.tex

@@ -76,7 +76,23 @@ $G$.
 \begin{definition}
 \label{def:level-edge}
 An edge $\{u, v\} \in E(G)$ is a \emph{level edge} if
-$\mathrm{depth}(u) = \mathrm{depth}(v)$.
+$\mathrm{depth}(u) = \mathrm{depth}(v)$, and an \emph{interlevel
+edge} if $\mathrm{depth}(u) \neq \mathrm{depth}(v)$.
+\end{definition}
+
+Under planar duality, each primal edge of $G$ corresponds to a
+unique edge of the planar dual --- the edge between the two face
+vertices on either side of the primal edge.  Levels and interlevels
+lift to the dual:
+
+\begin{definition}
+\label{def:level-dual-edge}
+A dual edge of $G$ is a \emph{level dual edge} if it crosses a level
+edge of $G$ in the planar embedding (equivalently, if its
+corresponding primal edge has equal-depth endpoints), and an
+\emph{interlevel dual edge} if it crosses an interlevel edge
+(equivalently, if its corresponding primal edge has
+different-depth endpoints).
 \end{definition}
 
 \begin{definition}