  Motivated by the planarization of 2-layered straight-line drawings, we
consider the problem of modifying a graph such that the resulting graph has
pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks
whether such a graph can be obtained using at most $k$ vertex explosions, where
a vertex explosion replaces a vertex $v$ by deg$(v)$ degree-1 vertices, each
incident to exactly one edge that was originally incident to $v$. For POVE, we
give an FPT algorithm with running time $O(4^k \cdot m)$ and an $O(k^2)$
kernel, thereby improving over the $O(k^6)$-kernel by Ahmed et al. [GD 22] in a
more general setting. Similarly, a vertex split replaces a vertex $v$ by two
distinct vertices $v_1$ and $v_2$ and distributes the edges originally incident
to $v$ arbitrarily to $v_1$ and $v_2$. Analogously to POVE, we define the
problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split
operation instead of vertex explosions. Here we obtain a linear kernel and an
algorithm with running time $O((6k+12)^k \cdot m)$. This answers an open
question by Ahmed et al. [GD22].
  Finally, we consider the problem $\Pi$ Vertex Splitting ($\Pi$-VS), which
generalizes the problem POVS and asks whether a given graph can be turned into
a graph of a specific graph class $\Pi$ using at most $k$ vertex splits. For
graph classes $\Pi$ that can be tested in monadic second-order graph logic
(MSO$_2$), we show that the problem $\Pi$-VS can be expressed as an MSO$_2$
formula, resulting in an FPT algorithm for $\Pi$-VS parameterized by $k$ if
$\Pi$ additionally has bounded treewidth. We obtain the same result for the
problem variant using vertex explosions.
