case of the directed Ising (parity-conserving) class (see Sec. \[Sec:discussion\]).

The PCPD [*per se*]{} can be considered one of crossover models from the DP${}_\textrm{IM}$ by allowing diffusion, which reduces the number of absorbing states considerably (from exponentially many to linearly many absorbing states with respect to system size). However, the crossover study from the PCP to the PCPD does not give any useful information on the DP${}_\textrm{IM}$, because the particle diffusion makes the system more active (excitatory) and the crossover turns out to be discontinuous.

Finally, we study the crossover from the diffusive reaction models to the DP. Such an example is the crossover from the PCPD to the DP studied in Ref. [@PP06]. In Sec. \[Sec:dptodp\], the hybrid-type models with diffusion ([tp12]{}: $2A\rightarrow\emptyset, 3A\rightarrow 4A$) [@KC03] are perturbed by adding a single-particle annihilation process ($A\rightarrow\emptyset$) and its crossover to the DP is investigated. These hybrid-type models (where the branching process is of higher order than the annihilation process) are numerically known to belong to the DP class. As expected, the critical line emanates “linearly” from the hybrid-type model point. We suggest an argument why these models should belong to the DP rather than a PCPD-type nontrivial class.

\[Sec:single\_to\_single\] Crossover 