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 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) center dot m) and an O(k (2)) kernel, thereby improving over the O(k(6))-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) center dot m). This answers an open question by Ahmed et al. [2].