Parameterized Complexity of Vertex Splitting to Pathwidth at most 1
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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 · m) and a quadratic 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 · m). This answers an open question by Ahmed et al. [GD22]. Finally, we consider the problem Π Vertex Splitting (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 tested in monadic second-order graph logic (MSO_2), we show that the problem Π-VS can be expressed as an MSO_2 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.
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