Nonplanar vertex deletion: maximum degree thresholds for NP/Max SNP-hardness and a -approximation for finding maximum planar induced subgraphs

The non planar vertex deletion v d ( G ) , of a graph G is the smallest positive integer k, such that the removal of k vertices from G produces a planar graph. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a positive integer k, to decide whether v d ( G ) ≤ k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a positive integer k satisfy v d ( G ) ≤ k . We prove that to compute v d ( G ) is Max SNP-hard when restricted to a cubic input G. We exhibit a polynomial 3 4 -approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.