\documentclass[tikz,border=5pt]{standalone}
\usepackage{amsmath,amssymb,amsthm,tikz}
\usetikzlibrary{decorations.pathmorphing,decorations.markings,graphs,quotes}
\begin{document}
\begin{tikzpicture}[scale=0.9, auto=left, v/.style={circle,draw=black,minimum size=12pt,inner sep=0.3pt}, l/.style={minimum size=24pt,inner sep=0.3pt}, b/.style={line width=0.5mm,blue}, r/.style={line width=0.5mm,red,dotted}, g/.style={line width=0.5mm,green,dash dot}]
\node[l] at (-1,0) {\large $T_{20}$ :};
\foreach \i/\x/\y in {1/0/0,2/1/0,3/2/0,4/3/0,5/4/0,6/2/1,7/3/1,8/3/-1}
\node[v] (n\i) at (\x,\y) {\i};
\path[b] (n1) edge node [above] {$\Gamma_4$} (n2) (n2) edge (n3) (n3) edge (n4) (n4) edge (n5) (n3) edge (n6) (n4) edge (n7) (n4) edge (n8);
\path[g] (n1) edge [bend right=90] node [below] {$\Gamma_2$} (n4);
\path[r] (n2) edge [bend right=90] node [above] {$\Gamma_3$} (n4) (n1) edge [bend right=90] node [below] {$\Gamma_3$} (n8);
\node[l] at (-1,-1.8) {\large $\downarrow$};
\begin{scope}[yshift=-3.5cm]
\node[l] at (-2,0) {\large Not cycle-free :};
\foreach \i/\x/\y in {1/0/0,2/1/0,3/2/0,4/3/0,5/4/0,6/2/1,7/3/1,8/3/-1}
\node[v] (n\i) at (\x,\y) {\i};
\path[r] (n1) edge node [above] {$\Delta_3$} (n2);
\path[b] (n2) edge (n3) (n3) edge (n4) (n4) edge (n5) (n3) edge (n6) (n4) edge (n7) (n4) edge (n8);
\path[g] (n1) edge [bend right=90] node [below] {$\Delta_2$} (n4);
\path[b] (n2) edge [bend right=90] node [above] {$\Delta_4$} (n4);
\path[r] (n1) edge [bend right=90] node [below] {$\Delta_3$} (n8);
\end{scope}\node[l] at (-1,-5.3) {or};
\begin{scope}[yshift=-7cm]
\node[l] at (-2,0) {\large Not cycle-free :};
\foreach \i/\x/\y in {1/0/0,2/1/0,3/2/0,4/3/0,5/4/0,6/2/1,7/3/1,8/3/-1}
\node[v] (n\i) at (\x,\y) {\i};
\path[g] (n1) edge node [above] {$\Delta_2$} (n2);
\path[b] (n2) edge (n3) (n3) edge (n4) (n4) edge (n5) (n3) edge (n6) (n4) edge (n7) (n4) edge (n8);
\path[r] (n1) edge [bend right=90] node [below] {$\Delta_3$} (n4);
\path[b] (n2) edge [bend right=90] node [above] {$\Delta_4$} (n4);
\path[r] (n1) edge [bend right=90] node [below] {$\Delta_3$} (n8);
\end{scope}\node[l] at (-1,-8.3) {or};
\begin{scope}[yshift=-10.5cm]
\node[l] at (-3.3,0) {\large In $\Delta_4$ vertex $4$ has degree $5$ :};
\foreach \i/\x/\y in {1/0/0,2/1/0,3/2/0,4/3/0,5/4/0,6/2/1,7/3/1,8/3/-1}
\node[v] (n\i) at (\x,\y) {\i};
\path[r] (n1) edge node [above] {$\Delta_3$} (n2);
\path[b] (n2) edge (n3) (n3) edge (n4) (n4) edge (n5) (n3) edge (n6) (n4) edge (n7) (n4) edge (n8);
\path[b] (n1) edge [bend right=90] node [below] {$\Delta_4$} (n4);
\path[g] (n2) edge [bend right=90] node [above] {$\Delta_2$} (n4);
\path[r] (n1) edge [bend right=90] node [below] {$\Delta_3$} (n8);
\end{scope}
\begin{scope}[yshift=-14cm]
\node[l] at (-3.3,0) {\large In $\Delta_4$ vertex $4$ has degree $5$ :};
\foreach \i/\x/\y in {1/0/0,2/1/0,3/2/0,4/3/0,5/4/0,6/2/1,7/3/1,8/3/-1}
\node[v] (n\i) at (\x,\y) {\i};
\path[g] (n1) edge node [above] {$\Delta_2$} (n2);
\path[b] (n2) edge (n3) (n3) edge (n4) (n4) edge (n5) (n3) edge (n6) (n4) edge (n7) (n4) edge (n8);
\path[b] (n1) edge [bend right=90] node [below] {$\Delta_4$} (n4);
\path[r] (n2) edge [bend right=90] node [above] {$\Delta_3$} (n4);
\path[r] (n1) edge [bend right=90] node [below] {$\Delta_3$} (n8);
\end{scope}
\begin{scope}[yshift=-17.5cm]
\node[l] at (-3,0) {\large Switch $(1,4)$ with $(2,4)$ :};
\foreach \i/\x/\y in {1/0/0,2/1/0,3/2/0,4/3/0,5/4/0,6/2/1,7/3/1,8/3/-1}
\node[v] (n\i) at (\x,\y) {\i};
\path[b] (n1) edge node [above] {$\Delta_4$} (n2) (n2) edge (n3) (n3) edge (n4) (n4) edge (n5) (n3) edge (n6) (n4) edge (n7) (n4) edge (n8);
\path[r] (n1) edge [bend right=90] node [below] {$\Delta_3$} (n4);
\path[g] (n2) edge [bend right=90] node [above] {$\Delta_2$} (n4);
\path[r] (n1) edge [bend right=90] node [below] {$\Delta_3$} (n8);
\end{scope}
\end{tikzpicture}
\end{document}