\node (a) at ( 0 ,  0 ) {};
\node (b) [right=of a] {};
\node (c) at ( 0 . 5 ,  1 ) {};
\node (d) [right=of c] {};

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{positioning}
\begin{document}

\begin{tikzpicture}
\tikzstyle{dot} = [fill,draw, circle, minimum size=2pt, inner sep=0pt]
\node[dot, label=left :$a$] (a) at ( 126 : 2 ) {};
\node[dot, label=right:$c$] (c) at  ( 54 : 2 ) {};
\node[dot, label=right:$d$] (d) at ( 342 : 2 ) {};
\node[dot, label=below:$f$] (f) at ( 270 : 2 ) {};
\node[dot, label=left :$h$] (h) at ( 198 : 2 ) {};
\node[dot, label=above:$b$] (b) at ($(a)!. 5 !(c)$) {};
\node[dot, label=below right:$e$] (e) at ($(d)!. 5 !(f)$) {};
\node[dot, label=below left :$g$] (g) at ($(f)!. 5 !(h)$) {};
\node[dot, label=below left :$i$] (i) at ($(h)!. 5 !(a)$) {};
\draw (a) -- (b) -- (c) -- (d) -- (e) -- (f) -- (g) -- (h) -- (i) -- (a);
\draw (a) -- (d) -- (h) -- (c) -- (f) -- (a);
\draw (i) .. controls +(- 1 ,  2 ) and +( 0 ,  1 ) .. (b);
\draw (a) .. controls +( 2 ,  2 ) and +( 1 . 5 ,  1 . 5 ) .. (d);
\end{tikzpicture}

\end{document}