  Eigenstate coalescence in non-Hermitian systems is widely observed in diverse
scientific domains encompassing optics and open quantum systems. Recent
investigations have revealed that adiabatic encircling of exceptional points
(EPs) leads to a nontrivial Berry phase in addition to an exchange of
eigenstates. Based on these phenomena, we propose in this work an exhaustive
classification framework for EPs in non-Hermitian physical systems. In contrast
to previous classifications that only incorporate the eigenstate exchange
effect, our proposed classification gives rise to finer $\mathbb{Z}_2$
classifications depending on the presence of a $\pi$ Berry phase after the
encircling of the EPs. Moreover, by mapping arbitrary one-dimensional systems
to the adiabatic encircling of EPs, we can classify one-dimensional
non-Hermitian systems characterized by topological phase transitions involving
EPs. Applying our exceptional classification to various one-dimensional models,
such as the non-reciprocal Su--Schrieffer--Heeger (SSH) model, we exhibit the
potential for enhancing the understanding of topological phases in
non-Hermitian systems. Additionally, we address exceptional bulk-boundary
correspondence and the emergence of distinct topological boundary modes in
non-Hermitian systems.
