On maximum planar induced subgraphs

Abstract

The nonplanar vertex deletion or vertex deletion $\mathit{vd}(G)$ of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph $G^{\prime }$. In this case $G^{\prime }$ is said to be a maximum planar induced subgraph of G. 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 nonnegative integer k, to decide whether $\mathit{vd}(G)⩽k$ is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy $\mathit{vd}(G)⩽k$. We prove that unless P=NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute $\mathit{vd}(G)$ when restricted to a cubic input G. Finally, we exhibit a polynomial-time $\frac{3}{4}$-approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.