The methodology of response reactions (RERs) introduced earlier from thermodynamic and kinetic considerations is used in this work to develop a new algorithm for the classification and enumeration of unique/direct reaction routes (RRs) and overall reactions (ORs). According to the RERs approach, a unique set of both RRs and ORs may be generated starting from any conceivable set of linearly independent RRs and ORs. In particular, the direct ORs may be most conveniently enumerated starting from the formula matrix of the terminal species (reactants and products), i.e., without any relation to the elementary reactions comprising the detailed mechanism. Depending on the type of ORs produced by the RRs one can distinguish between two distinct types of direct RRs. Namely, one option is to define a direct RR by specifying the intermediate species that need to be eliminated. This type of RR is referred to as Milner RRs. The other option is to require the direct RRs to produce RERs, thus resulting in RRs referred to as Happel-Sellers RRs.