didericis
47a87a4c2b
In `plane_depth_sequencing/paper.tex`:
- Lemma 2.6 now allows any nonempty source S ⊆ V(G) whose vertices all
lie on the boundary of the outer face of the chosen embedding,
rather than only the outer-cycle case S = V(C).
- The proof is the same argument with S in place of C: at d=0 each
S-vertex remains on the outer face after restriction; for d ≥ 1
the BFS ball V_{<d}^S is connected and reaches the outer face
via S.
- The original outer-cycle statement is preserved as a remark inside
the lemma.
- Adds \label{lem:outerplanarity}.
In `coloring_nested_tire_graphs/paper.tex`:
- The proof of Lemma 1.7 drops the "extends verbatim" caveat and
simply cites the generalised Lemma 2.6, noting that since the level
source S is a single vertex (per the local Level-source definition)
we may freely choose an embedding placing S on the outer face;
outerplanarity is a graph property so the conclusion transfers.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
17 lines
1.4 KiB
TeX
17 lines
1.4 KiB
TeX
\relax
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Motivation}}{1}{}\protected@file@percent }
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\citation{baker1994}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{2}{}\protected@file@percent }
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\newlabel{lem:outerplanarity}{{2.6}{2}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Quadrilateral sequencing}}{3}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The deep embedding $G'$ of a small maximal planar graph (drawn with one outer-cap face as the outer face), with each quadrilateral $Q_n$ of the canonical sequence labelled by its index and the move code (AD = anchor drop, LA = level add, J = join, RC = ring completion) of the move that produced it. Solid edges are non-level; dashed edges are level. Background colour encodes quadrilateral type: amber for shallow diamonds, teal for deep diamonds, pink for S quads. Outer-cycle vertices are blue, the outer-cap vertex $x^{*}$ is red. The move-code string for this example is $01211333$.}}{6}{}\protected@file@percent }
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\newlabel{fig:example-sequence}{{1}{6}}
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\bibcite{baker1994}{1}
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\gdef \@abspage@last{8}
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