Parameterized complexity of vertex splitting to pathwidth at most 1

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 $\mathrm{deg}\left(v\right)$ 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 $O\left(k^2\right)$ kernel, thereby improving over the $O\left(k^6\right)$-kernel by Ahmed et al. [2] 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. [2].

Finally, we consider the problem Π-VertexSplitting (Π-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class Π using at most k vertex splits. For graph classes Π that can be defined in monadic second-order graph logic (MSO₂), we show that the problem Π-VS can be expressed as an MSO₂ formula, resulting in an FPT algorithm for Π-VS parameterized by k if Π additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions.