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 routines presented in the literature, mostly in the context of claw‐free graphs. These reductions can be embedded in a simple algorithm that in at most |E(G)‖ steps builds a new graph G′ without proper and homogeneous pairs of cliques, and such that G and G′ agree on the value of some relevant invariant (or property).