Linear relations on star coefficients of the chromatic symmetric function

We prove that the coefficient of the star $\mathfrak{st}_{21^{n-2}}$ in the chromatic symmetric function $X_G$ determines whether a connected graph $G$ is $2$-connected. We also prove new linear relations on other star coefficients of chromatic symmetric functions. This allows us to find new bases for certain spans of chromatic symmetric functions. Finally, we relate the coefficient of the star $\mathfrak{st}_n$ to acyclic orientations.