Given a graph $G=(V,E)$ and an integer $k\in \mathbb{N}$, we investigate the 2-Eigenvalue Vertex Deletion (2-EVD) problem. The objective is to remove at most $k$ vertices such that the adjacency matrix of the resulting graph has at most two eigenvalues. It is established that the adjacency matrix of a graph has at most two eigenvalues if and only if the graph is a collection of equal-sized cliques. Thus, the 2-Eigenvalue Vertex Deletion amounts to removing a set of at most $k$ vertices to transform the graph into a collection of equal-sized cliques. The 2-Eigenvalue Edge Editing (2-EEE), 2-Eigenvalue Edge Deletion (2-EED) and 2-Eigenvalue Edge Addition (2-EEA) problems are defined analogously. We present a kernel of size $\mathcal{O}(k^{3})$ for $2$-EVD, along with an FPT algorithm with a running time of $\mathcal{O}^{*}(2^{k})$. For the problem $2$-EEE, we provide a kernel of size $\mathcal{O}(k^{2})$. Additionally, we present linear kernels of size $5k$ and $6k$ for $2$-EEA and $2$-EED respectively. For the $2$-EED, we also construct an algorithm with running time $\mathcal{O}^{*}(1.47^{k})$ . These results address open questions posed by Misra et al. (ISAAC 2023) regarding the complexity of these problems when parameterized by the solution size.

翻译：给定图 $G=(V,E)$ 与整数 $k\in \mathbb{N}$，本文研究 2-特征值顶点删除（2-EVD）问题。其目标在于删除至多 $k$ 个顶点，使得所得图的邻接矩阵至多具有两个特征值。已知图的邻接矩阵至多具有两个特征值当且仅当该图是一系列等规模团的集合。因此，2-特征值顶点删除问题等价于通过删除至多 $k$ 个顶点的集合将图转化为一系列等规模团的集合。类似地定义了 2-特征值边编辑（2-EEE）、2-特征值边删除（2-EED）与 2-特征值边添加（2-EEA）问题。我们为 $2$-EVD 提出了规模为 $\mathcal{O}(k^{3})$ 的核，并给出了运行时间为 $\mathcal{O}^{*}(2^{k})$ 的 FPT 算法。对于 $2$-EEE 问题，我们给出了规模为 $\mathcal{O}(k^{2})$ 的核。此外，我们为 $2$-EEA 和 $2$-EED 分别提出了规模为 $5k$ 和 $6k$ 的线性核。对于 $2$-EED，我们还构建了运行时间为 $\mathcal{O}^{*}(1.47^{k})$ 的算法。这些结果回答了 Misra 等人（ISAAC 2023）提出的关于这些问题在以解规模为参数时的复杂性的开放性问题。