We study a family of graph modification problems called the F-Vertex Splitting problem. Given a graph G, the task is to determine whether G can be transformed into a graph G-prime belonging to a graph class F through a sequence of at most k vertex splits. We investigate this problem for several target graph classes, namely constellations, cycle graphs, linear forests, and bipartite graphs. We analyze both inclusive and exclusive variants of vertex splitting, as introduced by Abu-Khzam and collaborators (ISCO 2018). Our results show that the F-Vertex Splitting problem is polynomial-time solvable when F is a cycle graph or a linear forest, for both variants. In contrast, when F is a constellation or a bipartite graph, the problem is NP-complete for both variants.

翻译：我们研究一类称为F-顶点分裂问题的图修改问题。给定一个图G，任务是判断是否可以通过最多k次顶点分裂操作，将G转换为属于图类F的图G'。我们针对多个目标图类（即星座图、环图、线性森林和二部图）探讨了该问题。我们分析了Abu-Khzam及其合作者（ISCO 2018）提出的顶点分裂的包容性与排他性变体。研究结果表明：当F为环图或线性森林时，两种变体的F-顶点分裂问题均可在多项式时间内求解；相反，当F为星座图或二部图时，两种变体的问题均属于NP完全问题。