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.

翻译：我们研究了Cograph删除问题的参数化复杂度，该问题要求判断是否可以通过删除图中至多$k$条边使其成为$P_4$自由图。这是一个著名的图修改问题，在计算生物学和社交网络分析中具有应用价值。现有的参数化算法均采用相似策略：寻找一个$P_4$结构，探索其局部结构以执行高效的递归分支。目前最优算法的时间复杂度为$O^*(2.303^k)$，由于分支情况的复杂性，该算法需要借助自动化搜索。鉴于进一步改进现有策略存在困难，我们设计了一种基于模分解的新方法。我们分别处理每个模及其商图，其中商图是核心问题。这将问题简化为在质图上求解，而质图中所有模均为平凡模。随后，我们利用Chudnovsky等人提出的特征定理：任何足够大的质图必然包含七种结构之一作为诱导子图。这些结构均包含大量$P_4$，且其数量随图规模线性增长。我们证明这些结构允许递归分支树算法在任意$ε> 0$条件下实现$O^*((2 + ε)^k)$的时间复杂度。这似乎是质图特征定理的首个算法应用，并可能适用于其他修改问题。为此，我们精确给出了$\H$自由编辑问题能够利用我们约化到质图方法的图类集合$\H$，这为改进其他修改问题的算法开辟了新途径。