For any fixed positive integer $r$ and a given budget $k$, the $r$-\textsc{Eigenvalue Vertex Deletion} ($r$-EVD) problem asks if a graph $G$ admits a subset $S$ of at most $k$ vertices such that the adjacency matrix of $G\setminus S$ has at most $r$ distinct eigenvalues. The edge deletion, edge addition, and edge editing variants are defined analogously. For $r = 1$, $r$-EVD is equivalent to the Vertex Cover problem. For $r = 2$, it turns out that $r$-EVD amounts to removing a subset $S$ of at most $k$ vertices so that $G\setminus S$ is a cluster graph where all connected components have the same size. We show that $r$-EVD is NP-complete even on bipartite graphs with maximum degree four for every fixed $r > 2$, and FPT when parameterized by the solution size and the maximum degree of the graph. We also establish several results for the special case when $r = 2$. For the vertex deletion variant, we show that $2$-EVD is NP-complete even on triangle-free and $3d$-regular graphs for any $d\geq 2$, and also NP-complete on $d$-regular graphs for any $d\geq 8$. The edge deletion, addition, and editing variants are all NP-complete for $r = 2$. The edge deletion problem admits a polynomial time algorithm if the input is a cluster graph, while the edge addition variant is hard even when the input is a cluster graph. We show that the edge addition variant has a quadratic kernel. The edge deletion and vertex deletion variants are FPT when parameterized by the solution size alone. Our main contribution is to develop the complexity landscape for the problem of modifying a graph with the aim of reducing the number of distinct eigenvalues in the spectrum of its adjacency matrix. It turns out that this captures, apart from Vertex Cover, also a natural variation of the problem of modifying to a cluster graph as a special case, which we believe may be of independent interest.

翻译：对于任意固定正整数 $r$ 和给定预算 $k$，$r$-\textsc{特征值顶点删除}（$r$-EVD）问题询问是否存在一个至多包含 $k$ 个顶点的子集 $S$，使得 $G\setminus S$ 的邻接矩阵至多有 $r$ 个不同的特征值。边删除、边添加和边编辑变体类似地定义。对于 $r = 1$，$r$-EVD 等价于顶点覆盖问题。对于 $r = 2$，$r$-EVD 相当于删除一个至多包含 $k$ 个顶点的子集 $S$，使得 $G\setminus S$ 是一个所有连通分量具有相同大小的簇图。我们证明了对于每个固定 $r > 2$，$r$-EVD 在最大度为 4 的二部图上也是 NP 完全的，并且在以解的大小和图的最大度为参数时是 FPT 的。我们还针对 $r = 2$ 的特殊情况建立了若干结果。对于顶点删除变体，我们证明了对于任何 $d\geq 2$，$2$-EVD 在无三角形和 $3d$-正则图上也是 NP 完全的，并且对于任何 $d\geq 8$ 在 $d$-正则图上也是 NP 完全的。对于 $r = 2$，边删除、添加和编辑变体都是 NP 完全的。如果输入是簇图，边删除问题允许多项式时间算法，而即使输入是簇图，边添加变体也是困难的。我们证明了边添加变体具有二次核。边删除和顶点删除变体在仅以解的大小为参数时是 FPT 的。我们的主要贡献是为通过修改图以降低其邻接矩阵谱中不同特征值数量的问题描绘了复杂性全景。结果表明，除了顶点覆盖问题之外，这还捕捉到了修改为簇图问题的一个自然变体作为特例，我们相信这可能具有独立的研究价值。