A fast algorithm to remove proper and homogeneous pairs of cliques (while preserving some graph invariants)

We introduce a family of reductions for removing proper and homogeneous pairs of cliques from a graph G. This family generalizes some reductions presented in the literature, mostly in the context of claw-free graphs. We show that these reductions can be embedded in a simple algorithm that in at most jE(G)j steps builds a new graph G0 such that G0 has no proper and homogeneous pairs of cliques, and G and G0 agree on the value of some relevant invariant (or property).