We study the parameterized complexity of the Cograph Deletion problem, which asks whether one can delete at most $k$ edges from a graph to make it $P_4$-free. This is a well-known graph modification problem with applications in computation biology and social network analysis. All current parameterized algorithms use a similar strategy, which is to find a $P_4$ and explore the local structure around it to perform an efficient recursive branching. The best known algorithm achieves running time $O^*(2.303^k)$ and requires an automated search of the branching cases due to their complexity. Since it appears difficult to further improve the current strategy, we devise a new approach using modular decompositions. We solve each module and the quotient graph independently, with the latter being the core problem. This reduces the problem to solving on a prime graph, in which all modules are trivial. We then use a characterization of Chudnovsky et al. stating that any large enough prime graph has one of seven structures as an induced subgraph. These all have many $P_4$s, with the quantity growing linearly with the graph size, and we show that these allow a recursive branch tree algorithm to achieve running time $O^*((2 + ε)^k)$ for any $ε> 0$. This appears to be the first algorithmic application of the prime graph characterization and it could be applicable to other modification problems. Towards this goal, we provide the exact set of graph classes $\H$ for which the $\H$-free editing problem can make use of our reduction to a prime graph, opening the door to improvements for other modification problems.

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