Let $X=left(
begin{array}{llll}
x_1 & ldots & x_{n-1}& x_n\
x_2& ldots & x_n & x_{n+1}
end{array}right)$ be the Hankel matrix of size $2times n$ and let $G$ be a closed graph on the vertex set $[n].$ We study the binomial ideal $I_Gsubset K[x_1,ldots,x_{n+1}]$ which is generated by all the $2$-minors of $X$ which correspond to the edges of $G.$ We show that $I_G$ is Cohen-Macaula...