Vertex splitting consists of taking a vertex $v$ in a graph and replacing it with two non-adjacent vertices whose combined neighborhoods is the neighborhood of $v$. The split is said to be exclusive when these neighborhoods are disjoint. In the Claw-Free (Exclusive) Vertex Splitting problem, we are given a graph $G$ and an integer $k$, and we are asked if we can perform at most $k$ (exclusive) vertex splits to obtain a claw-free graph. We consider the complexity of Claw-Free Exclusive Vertex Splitting and prove it to be NP-complete in general, while admitting a polynomial-time algorithm when the input graph has maximum degree 4. This result settles an open problem posed in [Firbas \& Sorge, ISAAC 2024]. We also show that our results can be generalized to $K_{1,c}$-Free Vertex Splitting for all $c \geq 3$.

翻译：顶点分割操作指在图中选取一个顶点$v$，将其替换为两个不相邻的新顶点，且这两个新顶点的邻域之并等于原顶点$v$的邻域。当这两个邻域互不相交时，该分割称为互斥分割。在爪形图无关（互斥）顶点分割问题中，给定图$G$与整数$k$，需判断是否可通过至多$k$次（互斥）顶点分割操作得到一个无爪形图的图。本文研究了爪形图无关互斥顶点分割问题的计算复杂性，证明了该问题在一般情况下是NP完全的，但在输入图最大度不超过4时存在多项式时间算法。这一结果解决了[Firbas \& Sorge, ISAAC 2024]中提出的一个开放性问题。同时，我们证明了该结果可推广至所有$c \geq 3$的$K_{1,c}$无关顶点分割问题。