Planarizing Graphs and Their Drawings by Vertex Splitting

Martin Nöllenburg, Manuel Sorge, Soeren Terziadis, Anaïs Villedieu, Hsiang-Yun Wu, Jules Wulms

Abstract: 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 \(v \in V\), introduces two new vertices \(v_1, v_2\), and distributes the edges formerly incident to v among \(v_1, v_2\). The splitting number problem is known to be NP-complete for abstract graphs and we provide a non-uniform fixed-parameter tractable (FPT) algorithm for this problem. We then shift focus to the splitting number of a given topological graph drawing in \(\mathbb {R}^2\), where the new vertices resulting from vertex splits must be re-embedded into the existing drawing of the remaining graph. We show NP-completeness of this embedded splitting number problem, 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.