Planarizing Graphs and their Drawings by Vertex Splitting
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The splitting number of a graph G=(V,E) is the minimum number of vertex splits required to turn G into a planar graph, where a vertex split removes a vertex v ∈ V, introduces two new vertices v_1, v_2, and distributes the edges formerly incident to v among its two split copies v_1, v_2. The splitting number problem is known to be NP-complete. In this paper we shift focus to the splitting number of graph drawings in ℝ^2, where the new vertices resulting from vertex splits can be re-embedded into the existing drawing of the remaining graph. We first provide a non-uniform fixed-parameter tractable (FPT) algorithm for the splitting number problem (without drawings). Then we show the NP-completeness of the splitting number problem for graph drawings, even for its two subproblems of (1) selecting a minimum subset of vertices to split and (2) for re-embedding a minimum number of copies of a given set of vertices. For the latter problem we present an FPT algorithm parameterized by the number of vertex splits. This algorithm reduces to a bounded outerplanarity case and uses an intricate dynamic program on a sphere-cut decomposition.
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