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PDF statistics: - 100 PDF objects out of 1000 (max. 8388607) - 60 compressed objects within 1 object stream + 116 PDF objects out of 1000 (max. 8388607) + 71 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 1 words of extra memory for PDF output out of 10000 (max. 10000000) + 13 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf index 541f83a..3e77a03 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf and b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex index 471836c..296800b 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex @@ -3,6 +3,8 @@ \usepackage{graphicx} \usepackage{geometry} \usepackage{booktabs} +\usepackage{tikz} +\usetikzlibrary{backgrounds,decorations.pathreplacing} \geometry{margin=1in} \title{The boundary cut tire $T_\partial$: closing the coverage gap} @@ -74,6 +76,56 @@ $T_\partial^{(i)}$ obtained from: \end{itemize} \end{defn} +\paragraph{Picture.} A thick-side example with $H_1$ a hexagonal +cycle in $G'_i$: the low-side face $f_\partial$ is the outer +(unbounded) region containing $3$ pendants (depth $0$); the +adjacent high-side face is inside the hexagon, containing depth-$2$ +edges. + +\begin{center} +\begin{tikzpicture}[scale=1.6] + \def\R{1.6} + \foreach \i in {0,...,5} { + \pgfmathsetmacro{\ang}{60*\i + 90} + \node[circle, fill=black, inner sep=1.3pt] (v\i) at (\ang:\R) {}; + } + \begin{scope}[on background layer] + \fill[red!8] (-3.2,-2.4) rectangle (3.2,2.4); + \fill[green!8] + (90:\R) -- (150:\R) -- (210:\R) -- (270:\R) -- + (330:\R) -- (30:\R) -- cycle; + \end{scope} + \foreach \i in {0,...,5} { + \pgfmathtruncatemacro{\j}{mod(\i+1,6)} + \draw[blue, very thick] (v\i) -- (v\j); + } + \foreach \i in {0,2,4} { + \pgfmathsetmacro{\ang}{60*\i + 90} + \coordinate (p\i) at (\ang:\R+0.85); + \draw[red, thick, dashed] (v\i) -- (p\i); + \node[circle, draw=red, fill=white, inner sep=1.2pt] at (p\i) {}; + } + \foreach \i in {1,3,5} { + \pgfmathsetmacro{\ang}{60*\i + 90} + \coordinate (q\i) at (\ang:\R-0.7); + \draw[green!55!black, thick, dashed] (v\i) -- (q\i); + \node[circle, fill=green!55!black, inner sep=1.2pt] at (q\i) {}; + } + \node[blue] at (0, \R+0.35) {\small cycle of $T_\partial$ ($= \partial f_\partial$, depth $1$)}; + \node[red] at (2.95, 1.55) {\small OUT pendants}; + \node[red] at (2.95, 1.25) {\small (depth $0$, cut)}; + \node[green!55!black] at (0, -0.05) {\small IN pendants (depth $2$)}; + \node at (-2.55, -1.95) {\small low-side face $f_\partial$}; + \node at (2.4, -2.05) {\small (high-side interior)}; +\end{tikzpicture} +\end{center} + +The cycle of $T_\partial$ is the hexagon (blue, depth $1$); pendants +fall in two classes: OUT (red, dashed, going into the low-side +face) attach to the cut, and IN (green, dashed, going into the +adjacent high-side region) attach to depth-$2$ edges shared with +$T_2^{(f')}$ tires. + \paragraph{What $T_\partial$ looks like in special cases.} \begin{itemize} \item \emph{Side with thick BFS:} $H_1$ has multiple faces, one @@ -89,6 +141,41 @@ $T_\partial^{(i)}$ obtained from: No IN pendants (no depth-$2$ edges exist). \end{itemize} +\begin{center} +\begin{tikzpicture}[scale=1.5] + \node[circle, fill=black, inner sep=1.3pt] (a) at (-1.5, 0) {}; + \node[circle, fill=black, inner sep=1.3pt] (b) at (-0.5, 0) {}; + \node[circle, fill=black, inner sep=1.3pt] (c) at ( 0.5, 0) {}; + \node[circle, fill=black, inner sep=1.3pt] (d) at ( 1.5, 0) {}; + \begin{scope}[on background layer] + \fill[red!8] (-3.2,-1.4) rectangle (3.2,1.4); + \end{scope} + \draw[blue, very thick] (a) -- (b) -- (c) -- (d); + \draw[blue, thick, decorate, decoration={brace, amplitude=4pt, raise=8pt}] + (a) -- (d) node[midway, above=12pt, blue] {\small $H_1$ (a tree, 3 edges)}; + \coordinate (pb) at (-0.5, -0.9); + \coordinate (pc) at ( 0.5, -0.9); + \draw[red, thick, dashed] (b) -- (pb); + \draw[red, thick, dashed] (c) -- (pc); + \node[circle, draw=red, fill=white, inner sep=1.2pt] at (pb) {}; + \node[circle, draw=red, fill=white, inner sep=1.2pt] at (pc) {}; + \node[red] at (-2.2, -0.5) {\small OUT pendants}; + \node[red] at (-2.2, -0.8) {\small (depth $0$)}; + \node at (2.5, 0.4) {\small low-side $f_\partial$}; + \node at (2.5, 0.1) {\small (the only face)}; + \draw[->, gray, thick] + (-1.5, 0.5) .. controls (-2.2, 1.2) and (2.2, 1.2) .. (1.5, 0.5); + \draw[->, gray, thick] + (1.5, -0.5) .. controls (2.2, -1.2) and (-2.2, -1.2) .. (-1.5, -0.5); + \node[gray] at (0, 1.05) {\small boundary walk traverses each edge twice}; +\end{tikzpicture} +\end{center} + +In the thin case (e.g.\ dodecahedron cut $\#0$ side $0$ with +$|S_0| = 4$), the boundary walk of the single face has length $6$ +(each of $3$ $H_1$ edges visited twice), and OUT pendants attach at +the two $V_{\deg = 2}$ vertices. + \section*{The extended forest} The high-side cut tire forest of $G'_i$ (proven in