Vertex splitting is a graph modification operation in which a vertex is replaced by multiple vertices such that the union of their neighborhoods equals the neighborhood of the original vertex. We introduce and study vertex splitting as a graph modification operation for transforming graphs into interval graphs. Given a graph $G$ and an integer $k$, we consider the problem of deciding whether $G$ can be transformed into an interval graph using at most $k$ vertex splits. We prove that this problem is NP-hard, even when the input is restricted to subcubic planar bipartite graphs. We further observe that vertex splitting differs fundamentally from vertex and edge deletions as graph modification operations when the objective is to obtain a chordal graph, even for graphs with maximum independent set size at most two. On the positive side, we give a polynomial-time algorithm for transforming, via a minimum number of vertex splits, a given graph into a disjoint union of paths, and that splitting triangle free graphs into unit interval graphs is also solvable in polynomial time.

翻译：顶点分裂是一种图修改操作，其中将一个顶点替换为多个顶点，使得这些新顶点邻域的并集等于原顶点的邻域。本文引入并研究了将顶点分裂作为将图转化为区间图的图修改操作。给定图$G$与整数$k$，我们考察判定$G$是否可通过至多$k$次顶点分裂转化为区间图的问题。我们证明该问题是NP难的，即使输入限制为次三次平面二部图亦然。我们进一步观察到，当目标为获得弦图时，顶点分裂作为一种图修改操作与顶点删除及边删除存在本质区别，即使对于最大独立集大小至多为二的图也是如此。在正面结果方面，我们给出了多项式时间算法，可通过最小次数的顶点分裂将给定图转化为不相交路径的并，且将无三角形图分裂为单位区间图的问题同样可在多项式时间内求解。