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Problema de superposición en el gráfico.

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Problema de superposición en el gráfico. 
\documentclass{article} 
\usepackage{animate} 
\usepackage{tikz}
\usepackage{tkz-graph}
\usetikzlibrary{arrows}
\usetikzlibrary{lindenmayersystems}
\usepackage[paperheight=20.2cm,paperwidth=16.2cm,bottom=-1cm,top=-1cm,left=-1.8cm,right=0cm]{geometry}
\begin{document}
\begin{tikzpicture}
\foreach \x in {31,91,...,391}{%
\draw  (\x:3cm) circle (5pt)[fill=black];
\draw  (\x:5cm) circle (5pt)[fill=black];
\draw  (\x:3cm) [line width=1pt]--  (\x+60:3cm);
\draw  (\x:3cm) [line width=1pt]--  (\x+120:3cm);
\draw  (\x:3cm) [line width=1pt]--  (\x+180:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x-60:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x+60:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x-120:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x+120:3cm);
\draw (\x:5cm)[line width=1pt]..controls(\x+120:11cm)..(\x+180:3cm);
%\draw  (\x:3cm) [line width=3pt] -- (\x+144:3cm);
%\draw  (\x:3cm) [line width=3pt] -- (\x:3cm);}
\draw(31:3cm)node[right=2pt]{\huge $v_6$};
\draw(91:3cm)node[above=2pt]{\huge $v_1$};
\draw(151:3cm)node[left=2pt]{\huge $v_2$};
\draw(211:3cm)node[left=2pt]{\huge $v_3$};
\draw(271:3cm)node[below=2pt]{\huge $v_4$};
\draw(331:3cm)node[right=2pt]{\huge $v_5$};
\draw(271:5cm)node[below=3.5cm]{\huge $S'(K_6)$};
\draw(31:5cm)node[right=2pt]{\huge $v_6'$};
\draw(91:5cm)node[above=2pt]{\huge $v_1'$};
\draw(151:5cm)node[left=2pt]{\huge $v_2'$};
\draw(211:5cm)node[left=2pt]{\huge $v_3'$};
\draw(271:5cm)node[below=2pt]{\huge $v_4'$};
\draw(331:5cm)node[right=2pt]{\huge $v_5'$};
\end{tikzpicture}     
\end{document}

Estoy dibujando el gráfico de arriba, en el que las etiquetas internas se superponen con los bordes. Si alguien puede dibujar superponiendo o si alguien puede dibujar el mismo gráfico de manera diferente, también sería útil para mí.

Respuesta1

Primero, hay un error en su código: ha comentado la llave que termina el cuerpo del foreachbucle.

Respecto a su problema de diseño: úselo controlscon dos puntos.

\draw (A) .. controls (a) and (b) .. (B);

La línea saldrá (A)hacia (a)y se acercará (B)desde la dirección de (b).

En su aplicación considere reemplazar

\draw (\x:5cm)[line width=1pt]..controls(\x+120:11cm)..(\x+180:3cm);

por

\draw (\x:5cm)[line width=1pt]..controls(\x+100:10cm) and (\x+120:11cm)..(\x+180:3cm);

ingrese la descripción de la imagen aquí

\documentclass{article} 
\usepackage{animate} 
\usepackage{tikz}
\usepackage{tkz-graph}
\usetikzlibrary{arrows}
\usetikzlibrary{lindenmayersystems}
\usepackage[paperheight=20.2cm,paperwidth=16.2cm,bottom=-1cm,top=-1cm,left=-1.8cm,right=0cm]{geometry}
\begin{document}
\begin{tikzpicture}
\foreach \x in {31,91,...,391}{%
\draw  (\x:3cm) circle (5pt)[fill=black];
\draw  (\x:5cm) circle (5pt)[fill=black];
\draw  (\x:3cm) [line width=1pt]--  (\x+60:3cm);
\draw  (\x:3cm) [line width=1pt]--  (\x+120:3cm);
\draw  (\x:3cm) [line width=1pt]--  (\x+180:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x-60:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x+60:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x-120:3cm);
\draw (\x:5cm) [line width=1pt]--  (\x+120:3cm);
%\draw (\x:5cm)[line width=1pt]..controls(\x+120:11cm)..(\x+180:3cm);
\draw (\x:5cm)[line width=1pt]..controls(\x+100:10cm) and (\x+120:11cm)..(\x+180:3cm);

%\draw  (\x:3cm) [line width=3pt] -- (\x+144:3cm);
%\draw  (\x:3cm) [line width=3pt] -- (\x:3cm);
}
\draw(31:3cm)node[right=2pt]{\huge $v_6$};
\draw(91:3cm)node[above=2pt]{\huge $v_1$};
\draw(151:3cm)node[left=2pt]{\huge $v_2$};
\draw(211:3cm)node[left=2pt]{\huge $v_3$};
\draw(271:3cm)node[below=2pt]{\huge $v_4$};
\draw(331:3cm)node[right=2pt]{\huge $v_5$};
\draw(271:5cm)node[below=3.5cm]{\huge $S'(K_6)$};
\draw(31:5cm)node[right=2pt]{\huge $v_6'$};
\draw(91:5cm)node[above=2pt]{\huge $v_1'$};
\draw(151:5cm)node[left=2pt]{\huge $v_2'$};
\draw(211:5cm)node[left=2pt]{\huge $v_3'$};
\draw(271:5cm)node[below=2pt]{\huge $v_4'$};
\draw(331:5cm)node[right=2pt]{\huge $v_5'$};
\end{tikzpicture}     
\end{document}

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