For a second order reaction, the correct on plot of $t_{\frac{1}{2}} vs \frac{1}{{\left[A\right]}_o}$is 

For a second-order reaction, the half-life is inversely related to the initial concentration of the reactant (A). For a second-order reaction each half-life is twice as long as the life span of the one before.

$t_{1/2}=\frac{1}{k{\left[A\right]}_o}$

Compared the above equation with y=mx+c, where value of c is zero. Hence, graph is: