  In 1987, A. Gy\'arf\'as in his paper ``Problems from the world surrounding
perfect graphs'' posed the problem of determining the smallest $\chi$-binding
function for $\mathcal{G}(F,\overline{F})$, when $\mathcal{G}(F)$ is
$\chi$-bounded. So far the problem has been attempted for only forest $F$ with
four or five vertices. In this paper, we address the case when $F=P_3\cup 2K_1$
and show that if $G$ is a $\{P_3\cup 2K_1,\overline{P_3\cup 2K_1}\}$-free graph
with $\omega(G)\neq 3$, then it admits $\omega(G)+1$ as a $\chi$-binding
function. Moreover, we also construct examples to show that this bound is tight
for all values of $\omega\neq 3$.

Combinatorics