Obtaining a Planar Graph by Vertex Deletion

TL;DR

This paper presents a simpler, quadratic-time algorithm for the k-Apex problem, which involves deleting vertices to make a graph planar, improving upon the complex existing solutions.

Contribution

The authors introduce a more straightforward algorithm with quadratic running time for the k-Apex problem, leveraging graph reduction and bounded treewidth techniques.

Findings

Quadratic time complexity achieved for the k-Apex problem

Simplified algorithm compared to previous complex proofs

Effective reduction techniques for planarization

Abstract

In the k-Apex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour, there is an O(n^3) time algorithm for every fixed value of k. However, the proof is extremely complicated and the constants hidden by the big-O notation are huge. Here we give a much simpler algorithm for this problem with quadratic running time, by iteratively reducing the input graph and then applying techniques for graphs of bounded treewidth.