Planarizing Graphs and Their Drawings by Vertex Splitting

The splitting number of a graph $$G=(V,E)$$ is the minimum vertex splits required to turn G into planar graph, where split removes $$v \in V$$ , introduces two new vertices $$v_1, v_2$$ and distributes edges formerly incident v among . problem known be NP-complete for abstract graphs we provide non-uniform fixed-parameter tractable (FPT) algorithm this problem. We then shift focus given topological drawing in $$\mathbb {R}^2$$ resulting from must re-embedded existing remaining graph. show NP-completeness embedded problem, even its subproblems (1) selecting subset (2) re-embedding copies set vertices. For latter present an FPT parameterized by splits. This reduces bounded outerplanarity case uses intricate dynamic program on sphere-cut decomposition.