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-rw-r--r--mgr.tex16
-rw-r--r--partition.pdfbin0 -> 6447 bytes
-rw-r--r--partition.pdf_tex58
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diff --git a/mgr.tex b/mgr.tex
index c26ae1a..4b4d025 100644
--- a/mgr.tex
+++ b/mgr.tex
@@ -869,6 +869,22 @@ singletons constructed according to the following rules:
were assigned to parts when its children were handled.
\end{enumerate}
+\begin{figure}
+ \centering
+ \def\svgwidth{\columnwidth}
+ \input{partition.pdf_tex}
+ \caption{
+ An LCA-partition of the tree with respect to the black vertices. The
+ dark grey vertices are added when computing the LCA closure, the light
+ grey vertices are the final vertices needed to create a minimal
+ partition ready set containing the black vertices. Part (a) is an
+ example of a singleton, part (b) of a subtree with a hole, and (c) of a
+ subtree.
+ }\label{partition-figure}
+\end{figure}
+
+See Figure \ref{partition-figure} for an example of a partitioned tree.
+
If $W$ is not parition ready, we will define the LCA partition with respect to
$W$ the partition with respect to the minimal partition ready set that contains
$W$. Skipping ahead, the LCA partition of the set of vertices relabeled in a
diff --git a/partition.pdf b/partition.pdf
new file mode 100644
index 0000000..953ca4c
--- /dev/null
+++ b/partition.pdf
Binary files differ
diff --git a/partition.pdf_tex b/partition.pdf_tex
new file mode 100644
index 0000000..b5de94f
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+++ b/partition.pdf_tex
@@ -0,0 +1,58 @@
+%% Creator: Inkscape 1.1.1 (3bf5ae0d25, 2021-09-20, custom), www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'partition.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%% \input{<filename>.pdf_tex}
+%% instead of
+%% \includegraphics{<filename>.pdf}
+%% To scale the image, write
+%% \def\svgwidth{<desired width>}
+%% \input{<filename>.pdf_tex}
+%% instead of
+%% \includegraphics[width=<desired width>]{<filename>.pdf}
+%%
+%% Images with a different path to the parent latex file can
+%% be accessed with the `import' package (which may need to be
+%% installed) using
+%% \usepackage{import}
+%% in the preamble, and then including the image with
+%% \import{<path to file>}{<filename>.pdf_tex}
+%% Alternatively, one can specify
+%% \graphicspath{{<path to file>/}}
+%%
+%% For more information, please see info/svg-inkscape on CTAN:
+%% http://tug.ctan.org/tex-archive/info/svg-inkscape
+%%
+\begingroup%
+ \makeatletter%
+ \providecommand\color[2][]{%
+ \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
+ \renewcommand\color[2][]{}%
+ }%
+ \providecommand\transparent[1]{%
+ \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
+ \renewcommand\transparent[1]{}%
+ }%
+ \providecommand\rotatebox[2]{#2}%
+ \newcommand*\fsize{\dimexpr\f@size pt\relax}%
+ \newcommand*\lineheight[1]{\fontsize{\fsize}{#1\fsize}\selectfont}%
+ \ifx\svgwidth\undefined%
+ \setlength{\unitlength}{439.05363307bp}%
+ \ifx\svgscale\undefined%
+ \relax%
+ \else%
+ \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
+ \fi%
+ \else%
+ \setlength{\unitlength}{\svgwidth}%
+ \fi%
+ \global\let\svgwidth\undefined%
+ \global\let\svgscale\undefined%
+ \makeatother%
+ \begin{picture}(1,0.52063675)%
+ \lineheight{1}%
+ \setlength\tabcolsep{0pt}%
+ \put(0,0){\includegraphics[width=\unitlength,page=1]{partition.pdf}}%
+ \end{picture}%
+\endgroup%