Techniques for the Cograph Editing Problem: Module Merge is equivalent to Editing P4s

TL;DR

This paper introduces a new perspective on cograph editing by showing that module merging is equivalent to editing induced P4s, leading to an exact algorithm and potential heuristics for this NP-hard problem.

Contribution

It establishes the equivalence between P4 editing and module merging in the context of cograph editing, enabling new algorithmic approaches.

Findings

New exact algorithm for cograph editing

Module merge operation is equivalent to P4 editing

Potential for improved heuristics based on this equivalence

Abstract

Cographs are graphs in which no four vertices induce a simple connected path $P_4$. Cograph editing is to find for a given graph $G = (V,E)$ a set of at most $k$ edge additions and deletions that transform $G$ into a cograph. This combinatorial optimization problem is NP-hard. It has, recently found applications in the context of phylogenetics, hence good heuristics are of practical importance. It is well-known that the cograph editing problem can be solved independently on the so-called strong prime modules of the modular decomposition of $G$. We show here that editing the induced $P_4$'s of a given graph is equivalent to resolving strong prime modules by means of a newly defined merge operation on the submodules. This observation leads to a new exact algorithm for the cograph editing problem that can be used as a starting point for the construction of novel heuristics.