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Simplify d=0 case in outerplanar lemma proof

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Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
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didericis
2026-04-25 05:22:25 -04:00
parent 5605e035d3
commit 927f9b7adb
5 changed files with 27 additions and 72 deletions
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papers/plane_depth_sequencing/paper.fdb_latexmk
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@@ -109,7 +109,7 @@ Let $G$ be a graph with a plane embedding and outer cycle $C$. For each $d \geq
\begin{proof} \begin{proof}
Let $H = G[V_d]$ with the plane embedding inherited from $G$. It suffices to show every vertex of $H$ lies on the outer face of $H$. Let $H = G[V_d]$ with the plane embedding inherited from $G$. It suffices to show every vertex of $H$ lies on the outer face of $H$.
For $d = 0$, we have $V_0 = V(C)$ and $H$ is a subgraph of the cycle $C$, hence a disjoint union of paths, which is outerplanar. For $d = 0$, we have $V_0 = C$, so $H$ is outerplanar.
For $d \geq 1$, let $U$ be the open subset of the plane obtained by removing all vertices and edges of $H$. We show every $v \in V_d$ lies on the boundary of the component $U_{\mathrm{out}}$ of $U$ containing the outer face of $G$. For $d \geq 1$, let $U$ be the open subset of the plane obtained by removing all vertices and edges of $H$. We show every $v \in V_d$ lies on the boundary of the component $U_{\mathrm{out}}$ of $U$ containing the outer face of $G$.