  Percolation is a paradigmatic model in disordered systems and has been
applied to various natural phenomena. The percolation transition is known as
one of the most robust continuous transitions. However, recent extensive
studies have revealed that a few models exhibit a discontinuous percolation
transition (DPT) in cluster merging processes. Unlike the case of continuous
transitions, understanding the nature of discontinuous phase transitions
requires a detailed study of the system at hand, which has not been undertaken
yet for DPTs. Here we examine the cluster size distribution immediately before
an abrupt increase in the order parameter of DPT models and find that DPTs
induced by cluster merging kinetics can be classified into two types. Moreover,
the type of DPT can be determined by the key characteristic of whether the
cluster kinetic rule is homogeneous with respect to the cluster sizes. We also
establish the necessary conditions for each type of DPT, which can be used
effectively when the discontinuity of the order parameter is ambiguous, as in
the explosive percolation model.
