diff --git a/lib/dual_graph.py b/lib/dual_graph.py deleted file mode 100644 index 6d2c97e..0000000 --- a/lib/dual_graph.py +++ /dev/null @@ -1,29 +0,0 @@ -"""Utilities for working with planar dual graphs.""" -from typing import Any -from sage.all import Graph - - -def find_vertex_for_dual_face(dual_face: Any) -> Any | None: - """Return the primal vertex shared by all faces in the dual face, or None.""" - shared_vertices = None - for dual_edge in dual_face: - vertices = set(map(lambda e: e[0], dual_edge[0])) - if shared_vertices: - shared_vertices.intersection_update(vertices) - else: - shared_vertices = vertices - if len(shared_vertices) == 1: - return shared_vertices.pop() - return None - - -def _dual_edge_has_vertex(dual_edge: Any, vertex: Any) -> bool: - return any(vertex in edge for dual_vertex in dual_edge for edge in dual_vertex) - - -def find_dual_face_for_removed_vertex(planar_dual: Graph, vertex: Any) -> Any | None: - """Return the dual face of length 5 whose every dual edge contains the given vertex.""" - for dual_face in list[Any](planar_dual.faces()): # type: ignore[call-arg] - if len(dual_face) == 5 and all(_dual_edge_has_vertex(de, vertex) for de in dual_face): - return dual_face - return None diff --git a/lib/edge_for_dual_edge.py b/lib/edge_for_dual_edge.py deleted file mode 100644 index 6d1c27c..0000000 --- a/lib/edge_for_dual_edge.py +++ /dev/null @@ -1,12 +0,0 @@ -"""Utilities for finding primal edges corresponding to dual edges.""" -from typing import Any, cast - -def get_edge_for_dual_edge(dual_edge: Any) -> tuple[int, int, None]: - """Return the primal edge shared by both faces of the given dual edge.""" - edges: list[set[Any]] = [] - for e in (dual_edge[0] + dual_edge[1]): - edge = set(e) - if edge in edges: - return cast(tuple[int, int, None], e) - edges.append(edge) - raise ValueError(f"Error finding edge for {dual_edge}") diff --git a/lib/edge_graph.py b/lib/edge_graph.py deleted file mode 100644 index 383f0ea..0000000 --- a/lib/edge_graph.py +++ /dev/null @@ -1,42 +0,0 @@ -"""Utilities for constructing edge graphs from planar graphs.""" -from typing import Any, cast -from sage.all import Graph - - -def get_edge_graph_pos(edge_graph: Graph, pos_src: dict[Any, Any]) -> dict[Any, tuple[Any, Any]]: - """Return a position dict for edge graph vertices, using midpoints of the source positions.""" - pos: dict[Any, tuple[Any, Any]] = {} - for e in cast(list[Any], edge_graph.vertices()): # type: ignore - pos[e] = ( - (pos_src[e[0]][0] + pos_src[e[1]][0]) / 2, - (pos_src[e[0]][1] + pos_src[e[1]][1]) / 2 - ) - return pos - - -def get_edge_graph(graph: Graph, set_pos: bool = True) -> Graph: - """Return the edge graph of the given graph, optionally setting vertex positions.""" - pos_src: dict[Any, Any] | None = None - pos: dict[Any, tuple[Any, Any]] = {} - if set_pos: - pos_src = cast(dict[Any, Any], graph.get_pos()) # type: ignore - - g = Graph() - for e in cast(list[Any], graph.edges()): # type: ignore - g.add_vertex(e) # type: ignore - if pos_src: - pos[e] = ( - (pos_src[e[0]][0] + pos_src[e[1]][0]) / 2, - (pos_src[e[0]][1] + pos_src[e[1]][1]) / 2 - ) - - for v in cast(list[Any], graph.vertices()): # type: ignore - incident_edges = cast(list[Any], graph.edges_incident(v)) # type: ignore - if len(incident_edges) == 1: - continue - for i, e1 in enumerate(incident_edges): - e2 = incident_edges[(i + 1) % len(incident_edges)] - g.add_edge(e1, e2) # type: ignore - if set_pos: - g.set_pos(pos) # type: ignore - return g diff --git a/lib/tait_coloring.py b/lib/tait_coloring.py deleted file mode 100644 index 92251b7..0000000 --- a/lib/tait_coloring.py +++ /dev/null @@ -1,18 +0,0 @@ -"""Utilities for computing Tait colorings from vertex colorings of the dual graph.""" -from typing import Any, cast -from sage.all import Graph -from lib.edge_for_dual_edge import get_edge_for_dual_edge - -def get_tait_coloring(planar_dual: Graph, coloring: dict[Any, int]) -> dict[Any, int]: - """Return a Tait (edge 3-coloring) from a vertex 4-coloring of the dual graph.""" - tait_coloring: dict[Any, int] = {} - for dual_edge in cast(list[Any], planar_dual.edges()): # type: ignore - edge = get_edge_for_dual_edge(dual_edge) - colors = {coloring[edge[0]], coloring[edge[1]]} - if colors in ({0, 1}, {2, 3}): - tait_coloring[dual_edge] = 0 - elif colors in ({1, 2}, {0, 3}): - tait_coloring[dual_edge] = 1 - else: - tait_coloring[dual_edge] = 2 - return tait_coloring diff --git a/lib/vertex_coloring.py b/lib/vertex_coloring.py deleted file mode 100644 index 586296d..0000000 --- a/lib/vertex_coloring.py +++ /dev/null @@ -1,11 +0,0 @@ -"""Utilities for verifying vertex colorings.""" -from typing import Any, cast -from sage.all import Graph - -def check_vertex_coloring(graph: Graph, vertex_coloring: dict[Any, int]) -> bool: - """Raise ValueError if any two adjacent vertices share a color.""" - for v in cast(list[Any], graph.vertices()): # type: ignore - for nv in cast(list[Any], graph.neighbors(v)): # type: ignore - if vertex_coloring[nv] == vertex_coloring[v]: - raise ValueError(f"Color {nv} equal to color {v}") - return True diff --git a/papers/plane_depth_labelling/paper.aux b/papers/plane_depth_labelling/paper.aux deleted file mode 100644 index 05a181d..0000000 --- a/papers/plane_depth_labelling/paper.aux +++ /dev/null @@ -1,7 +0,0 @@ -\relax -\newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{0pt} -\newlabel{tocindent1}{0pt} -\newlabel{tocindent2}{0pt} -\newlabel{tocindent3}{0pt} -\gdef \@abspage@last{1} diff --git a/papers/plane_depth_labelling/paper.fdb_latexmk b/papers/plane_depth_labelling/paper.fdb_latexmk deleted file mode 100644 index fafa945..0000000 --- a/papers/plane_depth_labelling/paper.fdb_latexmk +++ /dev/null @@ -1,46 +0,0 @@ -# Fdb version 3 -["pdflatex"] 1776296023 "/Users/didericis/Code/magnum-opus-4ct/plane_depth_labelling/paper.tex" "paper.pdf" "paper" 1776296023 - "/Users/didericis/Code/magnum-opus-4ct/plane_depth_labelling/paper.tex" 1776296022 1937 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a/papers/plane_depth_sequencing/paper.pdf b/papers/plane_depth_sequencing/paper.pdf new file mode 100644 index 0000000..034f9d3 Binary files /dev/null and b/papers/plane_depth_sequencing/paper.pdf differ diff --git a/papers/plane_depth_sequencing/paper.synctex.gz b/papers/plane_depth_sequencing/paper.synctex.gz new file mode 100644 index 0000000..1cc5a6e Binary files /dev/null and b/papers/plane_depth_sequencing/paper.synctex.gz differ diff --git a/papers/plane_depth_sequencing/paper.tex b/papers/plane_depth_sequencing/paper.tex new file mode 100644 index 0000000..2ec2ef4 --- /dev/null +++ b/papers/plane_depth_sequencing/paper.tex @@ -0,0 +1,119 @@ +%% filename: amsart-template.tex +%% version: 1.1 +%% date: 2014/07/24 +%% +%% American Mathematical Society +%% Technical Support +%% Publications Technical Group +%% 201 Charles Street +%% Providence, RI 02904 +%% USA +%% tel: (401) 455-4080 +%% (800) 321-4267 (USA and Canada only) +%% fax: (401) 331-3842 +%% email: tech-support@ams.org +%% +%% Copyright 2008-2010, 2014 American Mathematical Society. +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3c +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3c or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is the American Mathematical +%% Society. +%% +%% ==================================================================== + +% AMS-LaTeX v.2 template for use with amsart +% +% Remove any commented or uncommented macros you do not use. + +\documentclass{amsart} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{Plane Depth Sequencing} + +% Remove any unused author tags. + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + + +\subjclass[2010]{Primary } + +\keywords{} + +\date{} + +\dedicatory{} + +\begin{abstract} +\end{abstract} + +\maketitle + +\section{Definitions} + +\begin{definition} +Let $G$ be a graph with a plane embedding, and let $C$ be the outer cycle of that embedding. The \emph{plane depth} of a vertex $v \in V(G)$ relative to the embedding and $C$ is +\[ + \mathrm{depth}(v) = \min_{u \in V(C)} d(v, u), +\] +where $d(v, u)$ denotes the graph distance between $v$ and $u$ in $G$. +\end{definition} + +\begin{definition} +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 +\[ + \mathrm{depth}(u) = \mathrm{depth}(v) = \mathrm{depth}(w), +\] +add a new vertex $x$ to $G$ adjacent to each of $u$, $v$, and $w$. +\end{definition} + +\begin{lemma} +Let $G'$ be the deep embedding of a maximal planar graph $G$. For each face of $G'$, the plane depths of its three vertices are either $d, d+1, d+1$ or $d, d, d+1$ for some $d \geq 0$. +\end{lemma} + +\begin{proof} +We first establish that for any edge $\{p, q\}$ in $G$, the depths of $p$ and $q$ differ by at most $1$. Suppose for contradiction that $\mathrm{depth}(p) = d$ and $\mathrm{depth}(q) = d + n$ for some $n \geq 2$. Since $\mathrm{depth}(p) = d$, there exists a path of length $d$ from $p$ to some vertex of $C$. Prepending the edge $\{q, p\}$ gives a path of length $d + 1$ from $q$ to $C$, so $\mathrm{depth}(q) \leq d + 1 < d + n$, a contradiction. The case $\mathrm{depth}(q) = d - n$ is handled identically: there exists a path of length $d - n$ from $q$ to some vertex of $C$, and prepending the edge $\{p, q\}$ gives a path of length $d - n + 1 \leq d - 1 < d$ from $p$ to $C$, contradicting $\mathrm{depth}(p) = d$. + +Since $G$ is a triangulation, every interior face of $G$ is a triangle $\{u,v,w\}$ with all three pairs adjacent. By the above, each pair of vertices in a triangle differs in depth by at most $1$, so no triangle can contain vertices of depths $d$ and $d+2$ simultaneously. The possible depth patterns for a triangle in $G$ are therefore exactly $d,d,d$, or $d,d,d+1$, or $d,d+1,d+1$. + +We now consider each case under the deep embedding. + +\textit{Case 1: depths $d,d,d+1$ or $d,d+1,d+1$.} These triangles are not modified by the deep embedding, so they remain as faces of $G'$ with the stated depth patterns, satisfying the lemma. + +\textit{Case 2: depths $d,d,d$.} The deep embedding inserts a new vertex $x$ adjacent to $u$, $v$, and $w$, replacing the face $\{u,v,w\}$ with three new faces $\{u,v,x\}$, $\{v,w,x\}$, and $\{u,w,x\}$. It remains to determine the depth of $x$ in $G'$. Since $x$ is adjacent only to $u$, $v$, and $w$, every path in $G'$ from $x$ to $C$ must pass through one of them, so $x$ has strictly greater depth than $u$, $v$, and $w$. Each of the three new faces thus has depth pattern $d,d,d+1$, satisfying the lemma. + +Since every face of $G'$ falls into one of these cases, the result follows. +\end{proof} + +\end{document} + +%----------------------------------------------------------------------- +% End of amsart-template.tex +%----------------------------------------------------------------------- diff --git a/plane_depth_sequencing.py b/plane_depth_sequencing.py new file mode 100644 index 0000000..c878666 --- /dev/null +++ b/plane_depth_sequencing.py @@ -0,0 +1,65 @@ +"""Plane depth sequencing on maximal planar graphs.""" +import random +from typing import Any, TypedDict +from sage.all import Graph, graphs # type: ignore[attr-defined] # pylint: disable=no-name-in-module + + +class DeeplyEmbeddedGraph(TypedDict): + graph: Graph + outer_cycle: list[Any] + plane_depth_labelling: dict[Any, int] + deep_embedding: Graph + + +def get_plane_depth_labelling(g: Graph, outer_cycle: list[Any]) -> dict[Any, int]: + """Return the plane depth of each vertex relative to the given outer cycle.""" + # equivalent to the commented out naive implementation: + # return {v: min(g.distance(v, u) for u in outer_cycle) for v in g.vertices()} + source = max(g.vertices()) + 1 + g_prime = g.copy() + g_prime.add_vertex(source) + g_prime.add_edges([(source, v) for v in outer_cycle]) + distances = g_prime.breadth_first_search(source, report_distance=True) + return {v: d - 1 for v, d in distances if v != source} + + +def deep_embedding(g: Graph, outer_cycle: list[Any], plane_depth_labelling: dict[Any, int] | None = None) -> Graph: + """ + Return the deep embedding of g relative to outer_cycle. + + For every interior triangular face whose three vertices all have the same + plane depth, a new vertex is added adjacent to each vertex of that face. + """ + if plane_depth_labelling is None: + plane_depth_labelling = get_plane_depth_labelling(g, outer_cycle) + outer_vertices = set(outer_cycle) + + embedding = g.get_embedding() + if embedding is None: + g.is_planar(set_embedding=True) + embedding = g.get_embedding() + + g_prime = g.copy() + new_vertex = max(g.vertices()) + 1 + + for face in g.faces(embedding): + face_vertices = [u for u, v in face] + if set(face_vertices) == outer_vertices: + continue + if plane_depth_labelling[face_vertices[0]] == plane_depth_labelling[face_vertices[1]] == plane_depth_labelling[face_vertices[2]]: + g_prime.add_vertex(new_vertex) + g_prime.add_edges([(new_vertex, v) for v in face_vertices]) + new_vertex += 1 + + return g_prime + + +def generate_example(n: int) -> DeeplyEmbeddedGraph: + """Generate a random maximal planar graph of size n and return the triangulation, outer cycle, and deep embedding.""" + g = graphs.RandomTriangulation(n) + g.is_planar(set_embedding=True) + embedding = g.get_embedding() + faces = g.faces(embedding) + outer_cycle = [u for u, v in random.choice(faces)] + plane_depth_labelling = get_plane_depth_labelling(g, outer_cycle) + return DeeplyEmbeddedGraph(graph=g, outer_cycle=outer_cycle, plane_depth_labelling=plane_depth_labelling, deep_embedding=deep_embedding(g, outer_cycle, plane_depth_labelling))