  We present impelling evidence of topological phase transitions induced by
electron-phonon (e-ph) coupling in an $\alpha$-$T_3$ Haldane-Holstein model
that presents smooth tunability between graphene ($\alpha=0$) and a dice
lattice $(\alpha=1)$. The e-ph coupling has been incorporated via the
Lang-Firsov transformation which adequately captures the polaron physics in the
high frequency (anti-adiabatic) regime, and yields an effective Hamiltonian of
the system through zero phonon averaging at $T=0$. While exploring the
signature of the phase transition driven by polaron and its interplay with the
parameter $\alpha$, we identify two regions based on the values of $\alpha$,
namely, the low to intermediate range $(0 < \alpha \le 0.6)$ and larger values
of $\alpha~(0.6 < \alpha < 1)$ where the topological transitions show distinct
behaviour. There exists a single critical e-ph coupling strength for the
former, below which the system behaves as a topological insulator characterized
by edge modes, finite Chern number, and Hall conductivity, with all of them
vanishing above this value, and the system undergoes a spectral gap closing
transition. Further, the critical coupling strength depends upon $\alpha$. For
the latter case $(0.6 < \alpha < 1)$, the scenario is more interesting where
there are two critical values of the e-ph coupling at which
trivial-topological-trivial and topological-topological-trivial phase
transitions occur for $\alpha$ in the range $[0.6:1]$. Our studies on e-ph
coupling induced phase transitions show a significant difference with regard to
the well-known unique transition occurring at $\alpha = 0.5$ (or at $0.7$) in
the absence of the e-ph coupling, and thus underscore the importance of
interaction effects on the topological phase transitions.
