Given a graph $G=(V,E)$ and an integer $k\in \mathbb{N}$, we study {\sc 2-Eigenvalue Vertex Deletion} (2-EVD), where the goal is to remove at most $k$ vertices such that the adjacency matrix of the resulting graph has at most 2 eigenvalues. It is known that the adjacency matrix of a graph has at most 2 eigenvalues if and only if the graph is a collection of equal sized cliques. So {\sc 2-Eigenvalue Vertex Deletion} amounts to removing a set of at most $k$ vertices such that the resulting graph is a collection of equal sized cliques. The {\sc 2-Eigenvalue Edge Editing} (2-EEE), {\sc 2-Eigenvalue Edge Deletion} (2-EED) and {\sc 2-Eigenvalue Edge Addition} (2-EEA) problems are defined analogously. We provide a kernel of size $\mathcal{O}(k^{3})$ for {\sc $2$-EVD}. For the problems {\sc $2$-EEE} and {\sc $2$-EED}, we provide kernels of size $\mathcal{O}(k^{2})$. Finally, we provide a linear kernel of size $6k$ for {\sc $2$-EEA}. We thereby resolve three open questions listed by Misra et al. (ISAAC 2023) concerning the complexity of these problems parameterized by the solution size.

翻译：给定图$G=(V,E)$和整数$k\in \mathbb{N}$，我们研究**2-特征值顶点删除**（2-EVD）问题，其目标是最多删除$k$个顶点，使得结果图的邻接矩阵最多有2个特征值。已知图的邻接矩阵最多有2个特征值当且仅当该图是一组大小相等的团。因此，**2-特征值顶点删除**等价于删除一组最多$k$个顶点，使得结果图成为一组大小相等的团。类似地定义了**2-特征值边编辑**（2-EEE）、**2-特征值边删除**（2-EED）和**2-特征值边添加**（2-EEA）问题。我们为**2-EVD**问题提供了一个大小为$\mathcal{O}(k^{3})$的核，为**2-EEE**和**2-EED**问题提供了大小为$\mathcal{O}(k^{2})$的核，最后为**2-EEA**问题提供了一个大小为$6k$的线性核。由此，我们解决了Misra等人（ISAAC 2023）提出的关于这些问题参数化解规模复杂性的三个开放问题。