An Evolutionary Algorithm for Graph Planarisation by Vertex Deletion

A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The proposed algorithm aims to delete a minimum number of vertices to achieve its goal. The vertex deletion number of a graph G = (V,E) is the smallest integer k ≥ 0 such that there is an induced planar subgraph of G obtained by the removal of k vertices of G. Considering that the corresponding decision problem is NPcomplete and an approximation algorithm for graph planarisation by vertices deletion does not exist, this work proposes an evolutionary algorithm that uses a constructive heuristic algorithm to planarise a graph. This constructive heuristic has time complexity of O(n+m), where m = |V | and n = |E|, and it is based on the PQ-trees data structure and on the vertex deletion operation. The algorithm performance is verified by means