Splitting number is NP-complete

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k greater than or equal to 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex nu by two nonadjacent vertices nu(1) and nu(2), and attaches the neighbors of nu either to nu(1) or to nu(2). We prove that the SPLITTING NUMBER decision problem is NP-complete, even when restricted to cubic graphs. We obtain as a consequence that PLANAR SUBGRAPH remains NP-complete when restricted to cubic graphs. Note that NP-completeness for cubic graphs also implies NP-completeness for graphs not containing a subdivision of Ks as a subgraph.