Add level edge definition to plane depth sequencing paper
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
This commit is contained in:
@@ -53,8 +53,8 @@ def plot_colored_graph_to_data_uri(g: Graph, coloring: VertexColoring) -> str:
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return f"data:image/png;base64,{base64.b64encode(png_bytes).decode()}"
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def canonize_and_save_graph(g: Graph) -> Graph:
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"""Canonize g, save to data/graphs/<graph_id>/graph.sobj, and return the canonical graph."""
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def canonize_and_save_graph(g: Graph) -> tuple[Graph, Path]:
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"""Canonize g, save to data/graphs/<graph_id>/graph.sobj, and return the canonical graph and its folder."""
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canonical = cast(Graph, g.canonical_label())
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graph_id = base64.urlsafe_b64encode(canonical.graph6_string().encode()).decode()
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out_dir = DIR / "data" / "graphs" / graph_id
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@@ -63,7 +63,7 @@ def canonize_and_save_graph(g: Graph) -> Graph:
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canonical.is_planar(set_embedding=True, set_pos=True)
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canonical.plot().save(out_dir / 'graph.png')
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save(canonical, str(out_dir / 'graph'))
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return canonical
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return canonical, out_dir
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def save_colored_graph(g: Graph, coloring: VertexColoring) -> tuple[Graph, VertexColoring, ColoredGraphId]:
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@@ -1,6 +1,6 @@
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# Fdb version 4
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@@ -45,8 +45,8 @@
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(generated)
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"paper.aux"
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"paper.log"
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@@ -1,32 +1,43 @@
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This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 25 APR 2026 04:07
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entering extended mode
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restricted \write18 enabled.
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file:line:error style messages enabled.
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%&-line parsing enabled.
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**/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex
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(/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex
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**paper.tex
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(./paper.tex
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LaTeX2e <2025-06-01> patch level 1
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[2] (./paper.aux)
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***********
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@@ -86,6 +86,10 @@ Let $G$ be a graph with a plane embedding, and let $C$ be the outer cycle of tha
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where $d(v, u)$ denotes the graph distance between $v$ and $u$ in $G$.
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\end{definition}
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\begin{definition}
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An edge $\{u, v\} \in E(G)$ is a \emph{level edge} if $\mathrm{depth}(u) = \mathrm{depth}(v)$.
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\end{definition}
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\begin{definition}
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Let $G$ be a maximal planar graph with a plane embedding and outer cycle $C$. The \emph{deep embedding} of $G$ is the graph $G'$ obtained from $G$ by the following operation: for every 3-cycle $\{u, v, w\} \subseteq V(G)$ such that
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\[
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@@ -68,5 +68,6 @@ def generate_example(n: int) -> DeeplyEmbeddedGraph:
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if __name__ == "__main__":
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example = generate_example(10)
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canonical = canonize_and_save_graph(example['graph'])
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canonical, graph_dir = canonize_and_save_graph(example['graph'])
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(graph_dir / "plane_depth_sequence").mkdir(parents=True, exist_ok=True)
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print(canonical)
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