Abstract

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 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v 1 and v 2 , and attaches the neighbors of v either to v 1 or to v 2 . We prove that the splitting number decision problem is NP-complete. We obtain as a consequence that planar subgraph remains NP-complete when restricted to graphs with maximum degree 3, when restricted to graphs with no subdivision of K 5 , or when restricted to cubic graphs, problems that have been open since 1979.

Citations