A fast algorithm to remove proper and homogenous 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). 
Yuri Faenza, Gianpaolo Oriolo, and Claudia Snels
math.OC cs.DM math.CO