File:Nonisomorphic medial graph example.svg

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Description
English: Two embeddings of the same graph (in blue) can have nonisomorphic medial graphs (in red).
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\documentclass{article}
\thispagestyle{empty}
\usepackage[margin=0in]{geometry}
\usepackage{tikz}

\def\nodeDist{2.5cm}
\tikzstyle{origVertex}      = [draw, blue, fill, shape=circle]
\tikzstyle{newVertex}       = [draw, red,  fill, shape=circle]
\tikzstyle{invisibleVertex} = [shape=circle]
\tikzstyle{origEdge}      = [blue]
\tikzstyle{newEdge}       = [red, densely dashed]
\tikzstyle{invisibleEdge} = [draw opacity=0]

\begin{document}
\begin{tikzpicture}[node distance=\nodeDist,semithick]
 \node[origVertex] (0)              {};
 \node[origVertex] (1) [below of=0] {};
 \node[origVertex] (2) [left  of=1] {};
 \node[origVertex] (3) [right of=1] {};
 \node[origVertex] (4) [below of=1] {};
 \node[origVertex] (5) [below of=4] {};
 \path (0) edge[origEdge] node[invisibleVertex] (m0) {} (2)
           edge[origEdge] node[invisibleVertex] (m1) {} (3)
       (1) edge[origEdge] node[invisibleVertex] (m2) {} (4)
       (2) edge[origEdge] node[invisibleVertex] (m3) {} (4)
       (3) edge[origEdge] node[invisibleVertex] (m4) {} (4)
       (4) edge[origEdge] node[invisibleVertex] (m5) {} (5);
 \path (m0) edge[invisibleEdge, out=  90, in=  90, looseness=2]  (m1)
            edge[invisibleEdge]                                  (m1)
            edge[invisibleEdge, out= 180, in= 180, looseness=2]  (m3)
            edge[invisibleEdge]                                  (m3)
       (m1) edge[invisibleEdge, out=   0, in=   0, looseness=2]  (m4)
            edge[invisibleEdge]                                  (m4)
       (m2) edge[invisibleEdge, out= 125, in=  55, looseness=30] (m2)
            edge[invisibleEdge]                                  (m3)
            edge[invisibleEdge]                                  (m4)
       (m3) edge[invisibleEdge, bend right]                      (m5)
       (m4) edge[invisibleEdge, bend left]                       (m5)
       (m5) edge[invisibleEdge, out=-125, in= -55, looseness=30] (m5);
\end{tikzpicture}
\begin{tikzpicture}[node distance=\nodeDist,semithick]
 \node[origVertex] (0)              {};
 \node[origVertex] (1) [below of=0] {};
 \node[origVertex] (2) [left  of=1] {};
 \node[origVertex] (3) [right of=1] {};
 \node[origVertex] (4) [below of=1] {};
 \node[origVertex] (5) [below of=4] {};
 \path (0) edge[origEdge] node[newVertex] (m0) {} (2)
           edge[origEdge] node[newVertex] (m1) {} (3)
       (1) edge[origEdge] node[newVertex] (m2) {} (4)
       (2) edge[origEdge] node[newVertex] (m3) {} (4)
       (3) edge[origEdge] node[newVertex] (m4) {} (4)
       (4) edge[origEdge] node[newVertex] (m5) {} (5);
 \path (m0) edge[newEdge, out=  90, in=  90, looseness=2]  (m1)
            edge[newEdge]                                  (m1)
            edge[newEdge, out= 180, in= 180, looseness=2]  (m3)
            edge[newEdge]                                  (m3)
       (m1) edge[newEdge, out=   0, in=   0, looseness=2]  (m4)
            edge[newEdge]                                  (m4)
       (m2) edge[newEdge, out= 125, in=  55, looseness=30] (m2)
            edge[newEdge]                                  (m3)
            edge[newEdge]                                  (m4)
       (m3) edge[newEdge, bend right]                      (m5)
       (m4) edge[newEdge, bend left]                       (m5)
       (m5) edge[newEdge, out=-125, in= -55, looseness=30] (m5);
\end{tikzpicture}
\begin{tikzpicture}[node distance=\nodeDist,semithick]
 \node[invisibleVertex] (0)              {};
 \node[invisibleVertex] (1) [below of=0] {};
 \node[invisibleVertex] (2) [left  of=1] {};
 \node[invisibleVertex] (3) [right of=1] {};
 \node[invisibleVertex] (4) [below of=1] {};
 \node[invisibleVertex] (5) [below of=4] {};
 \path (0) edge[invisibleEdge] node[newVertex] (m0) {} (2)
           edge[invisibleEdge] node[newVertex] (m1) {} (3)
       (1) edge[invisibleEdge] node[newVertex] (m2) {} (4)
       (2) edge[invisibleEdge] node[newVertex] (m3) {} (4)
       (3) edge[invisibleEdge] node[newVertex] (m4) {} (4)
       (4) edge[invisibleEdge] node[newVertex] (m5) {} (5);
 \path (m0) edge[newEdge, out=  90, in=  90, looseness=2]  (m1)
            edge[newEdge]                                  (m1)
            edge[newEdge, out= 180, in= 180, looseness=2]  (m3)
            edge[newEdge]                                  (m3)
       (m1) edge[newEdge, out=   0, in=   0, looseness=2]  (m4)
            edge[newEdge]                                  (m4)
       (m2) edge[newEdge, out= 125, in=  55, looseness=30] (m2)
            edge[newEdge]                                  (m3)
            edge[newEdge]                                  (m4)
       (m3) edge[newEdge, bend right]                      (m5)
       (m4) edge[newEdge, bend left]                       (m5)
       (m5) edge[newEdge, out=-125, in= -55, looseness=30] (m5);
\end{tikzpicture}

\begin{tikzpicture}[node distance=\nodeDist,semithick]
 \node[origVertex]      (0)                    {};
 \node[invisibleVertex] (1) [below       of=0] {};
 \node[origVertex]      (2) [left        of=1] {};
 \node[origVertex]      (3) [right       of=1] {};
 \node[origVertex]      (4) [below       of=1] {};
 \node[origVertex]      (5) [below left  of=4] {};
 \node[origVertex]      (6) [below right of=4] {};
 \path (0) edge[origEdge] node[invisibleVertex] (m0) {} (2)
           edge[origEdge] node[invisibleVertex] (m1) {} (3)
       (2) edge[origEdge] node[invisibleVertex] (m2) {} (4)
       (3) edge[origEdge] node[invisibleVertex] (m3) {} (4)
       (4) edge[origEdge] node[invisibleVertex] (m4) {} (5)
           edge[origEdge] node[invisibleVertex] (m5) {} (6);
 \path (m0) edge[invisibleEdge, out=  90, in=  90, looseness=2]  (m1)
            edge[invisibleEdge]                                  (m1)
            edge[invisibleEdge, out= 180, in= 180, looseness=2]  (m2)
            edge[invisibleEdge]                                  (m2)
       (m1) edge[invisibleEdge, out=   0, in=   0, looseness=2]  (m3)
            edge[invisibleEdge]                                  (m3)
       (m2) edge[invisibleEdge]                                  (m3)
            edge[invisibleEdge]                                  (m4)
       (m3) edge[invisibleEdge]                                  (m5)
       (m4) edge[invisibleEdge, out=-170, in=-100, looseness=30] (m4)
            edge[invisibleEdge]                                  (m5)
       (m5) edge[invisibleEdge, out= -80, in= -10, looseness=30] (m5);
\end{tikzpicture}
\begin{tikzpicture}[node distance=\nodeDist,semithick]
 \node[origVertex]      (0)                    {};
 \node[invisibleVertex] (1) [below       of=0] {};
 \node[origVertex]      (2) [left        of=1] {};
 \node[origVertex]      (3) [right       of=1] {};
 \node[origVertex]      (4) [below       of=1] {};
 \node[origVertex]      (5) [below left  of=4] {};
 \node[origVertex]      (6) [below right of=4] {};
 \path (0) edge[origEdge] node[newVertex] (m0) {} (2)
           edge[origEdge] node[newVertex] (m1) {} (3)
       (2) edge[origEdge] node[newVertex] (m2) {} (4)
       (3) edge[origEdge] node[newVertex] (m3) {} (4)
       (4) edge[origEdge] node[newVertex] (m4) {} (5)
           edge[origEdge] node[newVertex] (m5) {} (6);
 \path (m0) edge[newEdge, out=  90, in=  90, looseness=2]  (m1)
            edge[newEdge]                                  (m1)
            edge[newEdge, out= 180, in= 180, looseness=2]  (m2)
            edge[newEdge]                                  (m2)
       (m1) edge[newEdge, out=   0, in=   0, looseness=2]  (m3)
            edge[newEdge]                                  (m3)
       (m2) edge[newEdge]                                  (m3)
            edge[newEdge]                                  (m4)
       (m3) edge[newEdge]                                  (m5)
       (m4) edge[newEdge, out=-170, in=-100, looseness=30] (m4)
            edge[newEdge]                                  (m5)
       (m5) edge[newEdge, out= -80, in= -10, looseness=30] (m5);
\end{tikzpicture}
\begin{tikzpicture}[node distance=\nodeDist,semithick]
 \node[invisibleVertex] (0)                    {};
 \node[invisibleVertex] (1) [below       of=0] {};
 \node[invisibleVertex] (2) [left        of=1] {};
 \node[invisibleVertex] (3) [right       of=1] {};
 \node[invisibleVertex] (4) [below       of=1] {};
 \node[invisibleVertex] (5) [below left  of=4] {};
 \node[invisibleVertex] (6) [below right of=4] {};
 \path (0) edge[invisibleEdge] node[newVertex] (m0) {} (2)
           edge[invisibleEdge] node[newVertex] (m1) {} (3)
       (2) edge[invisibleEdge] node[newVertex] (m2) {} (4)
       (3) edge[invisibleEdge] node[newVertex] (m3) {} (4)
       (4) edge[invisibleEdge] node[newVertex] (m4) {} (5)
           edge[invisibleEdge] node[newVertex] (m5) {} (6);
 \path (m0) edge[newEdge, out=  90, in=  90, looseness=2]  (m1)
            edge[newEdge]                                  (m1)
            edge[newEdge, out= 180, in= 180, looseness=2]  (m2)
            edge[newEdge]                                  (m2)
       (m1) edge[newEdge, out=   0, in=   0, looseness=2]  (m3)
            edge[newEdge]                                  (m3)
       (m2) edge[newEdge]                                  (m3)
            edge[newEdge]                                  (m4)
       (m3) edge[newEdge]                                  (m5)
       (m4) edge[newEdge, out=-170, in=-100, looseness=30] (m4)
            edge[newEdge]                                  (m5)
       (m5) edge[newEdge, out= -80, in= -10, looseness=30] (m5);
\end{tikzpicture}
\end{document}
Author Self

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I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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Date/TimeThumbnailDimensionsUserComment
current15:20, 9 December 2012Thumbnail for version as of 15:20, 9 December 2012699 × 598 (55 KB)Bender2k14 (talk | contribs){{Information |Description ={{en|1=Two embeddings of the same graph (in blue) can have nonisomorphic medial graphs (in red).}} |Source =<source lang="latex"> \documentclass{article} \thispagestyle{empty} \usepackage[margin=0in]{geometry} \us...

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