%\newcommand\PruferShort[4]{
%	\coordinate (x) at (#1,#2);
%	\fill (x) circle (.1);
%	\draw[#4] (x) .. controls (#1,#2+4) and (#1+.1,#2+4) .. (#1+#3,#2+3.8);
%}

\begin{tikzpicture}[scale=0.95]
\tikzstyle{every node} = [font = \footnotesize]
\node (a) at (0,0) [fill,violet,circle,inner sep=1pt] {};
\draw (0,0) circle (1.25cm);
\foreach \x in {-144,-72,0,72,144} {
   \begin{scope}[rotate=\x]
    \node (\x) at (0,-1.25) [fill,circle,inner sep=1pt] {};
   \end{scope}
}
%\draw (a) node [above] {$0$};
\draw (-72) node [left] {\tiny$v_2$};
\draw (0) node [below] {\tiny$v_3$};
\draw (72) node [right] {\tiny$v_4$};
\draw (144) node [above right] {\tiny{$v_5$}};
\draw (-144) node [above left] {\tiny$v_1$};

\draw (15:1.25cm) node [right] {$b_{45}$};
\draw (90:1.25cm) node [above] {$b_{51}$};
\draw (160:1.25cm) node [left] {$b_{12}$};
\draw (230:1.25cm) node [left] {$b_{23}$};
\draw (-55:1.25cm) node [right] {$b_{34}$};

\draw[thin] (a) to (-144);
%\draw[thin,red] (a) to (72);

\draw[thin,%opacity=0.5,
out=-140,in=-80] (0.2,-0.2) to (-144);
\draw[thin,%opacity=0.5,
out=-30,in=40] (-144) to  (0.2,-0.2);

%\draw[thin,opacity=0.5,out=-140,in=-80] (0.2,-0.2) to (-144);
%\draw[thin,opacity=0.5,out=-30,in=40] (-144) to  (0.2,-0.2);
\draw[thin,out=110,in=0] (72) to  (-144);
\draw[thin,out=-110,in=130] (-144) to (0);
\draw[thin,out=-170,in=-45] (72) to (-0.3,-0.3) ;
\draw[thin,out=135,in=-90] (-0.3,-0.3) to(-144);

\draw[red,rounded corners=6pt,densely dotted,->,very thick] (144) to (-0.2,0.1) to (0,-0.2) to (0.2,0.2) to (-0.4,0.3) to (-0.1,-0.3) to (72);

\fill(-.3,0.65) node[] {{$1$}}; %\tau
\fill(-.7,-.6) node[] {{$2$}}; %\tau
\fill(.2,-.7) node[] {{$3$}}; %\tau
\fill(.5,0) node[] {{$4$}}; %\tau
\fill(.85,.5) node[] {{$0$}}; %\tau

\fill(2.2,0) node[] {$\rightsquigarrow$};
\end{tikzpicture}
%\quad
		\begin{tikzpicture}[scale = .5,xscale=0.4]
		\tikzstyle{every node} = [font = \small]
		\foreach \x in {0,10,20}
		{
			\foreach \y in {-8}
			{
				\draw[] (\x-7,\y-3) -- (\x+6,\y-3);
			
				\foreach \t in {-6,-4,-2,0,2,4}
				{
					\fill (\x+\t,\y-3) circle (.1);
				}

				\fill (\x-6,\y-3) node [below] {\tiny$v_1$};
				\fill (\x-4,\y-3) node [below] {\tiny$v_2$};
				\fill (\x-2,\y-3) node [below] {\tiny$v_3$};
				\fill (\x+0,\y-3) node [below] {\tiny$v_4$};
				\fill (\x+2,\y-3) node [below] {\tiny$v_5$};
				\fill (\x+4,\y-3) node [below] {\tiny$v_1$};
				
				\PruferShort{\x-6}{\y-3}{13}{}
				\fill (\x-5.7,\y-1) node [left] {\tiny{$\tau_1$}};
				\draw[] (\x-6,\y-3) .. controls (\x-5,\y-2) and (\x-3,\y-2) .. (\x-2,\y-3);
				\fill (\x-2.3,\y-2.9) node [above] {\tiny{$\tau_2$}};
                \draw[] (\x-6,\y-3) .. controls (\x-4.5,\y) and (\x+3,\y) .. (\x+4,\y-3);
                \fill (\x-0,\y-0.5) node [right] {\tiny{$\tau_4$}};
				\draw[] (\x-6,\y-3) .. controls (\x-4.5,\y-1.25) and (\x-1.5,\y-1.25) .. (\x-0,\y-3);
				\fill (\x-3,\y-1.8) node [above] {\tiny{$\tau_3$}};

				\draw[] (\x+0,\y-3) .. controls (\x+1,\y-2) and (\x+3,\y-2) .. (\x+4,\y-3);
				\fill (\x+2,\y-2.4) node [above] {\tiny{$\tau_0$}};	
			}
		}	
        \PruferShort{20+4}{-8-3}{1}{}
        \draw[->] (20-7,-8-3) -- (20+6,-8-3);
        \fill (20+4,-8-1) node [right,black] {\tiny{$\tau_1$}};
        
        \foreach \x in {0}
		{
			\foreach \y in {-8}
			{
				\draw[red,densely dotted,->,very thick] (\x+2,\y-3) .. controls (\x+4,\y+1) and (\x+19,\y+1) .. (\x+20,\y-3);
              \fill (\x+17,\y) node[red]{$\gamma$};

                }
         }
\end{tikzpicture}