  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.
