Vertex splitting is a graph operation that replaces a vertex $v$ with two nonadjacent new vertices and makes each neighbor of $v$ adjacent with one or both of the introduced vertices. Vertex splitting has been used in contexts from circuit design to statistical analysis. In this work, we explore the computational complexity of achieving a given graph property $\Pi$ by a limited number of vertex splits, formalized as the problem $\Pi$ Vertex Splitting ($\Pi$-VS). We focus on hereditary graph properties and contribute four groups of results: First, we classify the classical complexity of $\Pi$-VS for graph properties characterized by forbidden subgraphs of size at most 3. Second, we provide a framework that allows to show NP-completeness whenever one can construct a combination of a forbidden subgraph and prescribed vertex splits that satisfy certain conditions. Leveraging this framework we show NP-completeness when $\Pi$ is characterized by forbidden subgraphs that are sufficiently well connected. In particular, we show that $F$-Free-VS is NP-complete for each biconnected graph $F$. Third, we study infinite families of forbidden subgraphs, obtaining NP-hardness for Bipartite-VS and Perfect-VS. Finally, we touch upon the parameterized complexity of $\Pi$-VS with respect to the number of allowed splits, showing para-NP-hardness for $K_3$-Free-VS and deriving an XP-algorithm when each vertex is only allowed to be split at most once.

翻译：顶点分裂是一种图操作，它将一个顶点 $v$ 替换为两个不相邻的新顶点，并使 $v$ 的每个邻点与引入的一个或两个顶点相邻。顶点分裂已应用于从电路设计到统计分析等多个领域。本文探讨了通过有限次顶点分裂实现给定图性质 $\Pi$ 的计算复杂性，并将其形式化为问题 $\Pi$ 顶点分裂 ($\Pi$-VS)。我们聚焦于遗传图性质，并贡献了四组结果：首先，针对由大小不超过3的禁止子图所刻画的图性质，我们分类了 $\Pi$-VS 的经典复杂性。其次，我们提出一个框架，该框架允许在能够构造满足特定条件的禁止子图与规定顶点分裂组合时，证明 NP-完全性。利用该框架，我们证明了当 $\Pi$ 由足够连通的禁止子图刻画时，其 NP-完全性；特别地，我们证明了对于每个双连通图 $F$，$F$-Free-VS 是 NP-完全的。第三，我们研究了无穷禁止子图族，获得了 Bipartite-VS 和 Perfect-VS 的 NP-困难性。最后，我们探讨了 $\Pi$-VS 关于允许分裂次数的参数化复杂性，证明了 $K_3$-Free-VS 的 para-NP-困难性，并在每个顶点至多允许被分裂一次时推导出一个 XP-算法。