Given an undirected graph $G=(V,E)$, a vertex $k$-cut of $G$ is a vertex subset of $V$ the removing of which disconnects the graph in at least $k$ components. Given a graph $G$ and an integer $k\ge 2$, the vertex $k$-cut problem consists in finding a vertex $k$-cut of $G$ of minimum cardinality. We first prove that the problem is NP-hard for any fixed $k\ge 3$. We then present a compact formulation, and an extended formulation from which we derive a column generation and a branching scheme. Extensive computational results prove the effectiveness of the proposed methods.

中文翻译：

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顶点 $ķ$切问题

给定无向图 $G=（V，Ë）$，一个顶点 $ķ$-切 $G$ 是的顶点子集 $V$ 删除至少会断开图形 $ķ$ 组件。给定图$G$ 和一个整数 $ķ\ge 2$，顶点 $ķ$割问题在于找到顶点 $ķ$-切 $G$最小基数。我们首先证明该问题对于任何固定问题都是NP困难的$ķ\ge 3$。然后，我们介绍了一个紧凑的公式和一个扩展的公式，从中可以得出色谱柱的产生和分支方案。大量的计算结果证明了所提方法的有效性。