Planarizing Graphs and Their Drawings by Vertex Splitting

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∈ V, introduces two new vertices v1, v2, and distributes the edges formerly incident to v among v1, v2. 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 R2, 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.