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

TL;DR

This paper presents a new, efficient algorithm for removing specific clique pairs from graphs while preserving key invariants, generalizing existing methods especially for claw-free graphs.

Contribution

It introduces a family of reductions that generalize previous routines, enabling a simple, linear-step algorithm to eliminate proper and homogeneous clique pairs while maintaining important graph properties.

Findings

Algorithm operates in at most |E(G)| steps

Successfully removes proper and homogeneous clique pairs

Preserves key graph invariants

Abstract

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).