Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with on-line algorithms.

A graph is l-apex if it can be made planar by removing at most l vertices. In this paper we show that the vertex set of any graph not containing an l- apex graph as a minor can be quickly partitioned into 2 l sets inducing graphs with small treewidth. As a consequence, several maximum induced-subgraph problems when restricted to graph classes not containing some special l-apex graphs as minors, have practical approximation algorithms. Keywords: Algorithms, Analysis of Algorithms, Approximation Algorithms, Combinatorial Problems, Graph Minors, Treewidth 1 Introduction Much work in algorithmic graph theory has been done in finding polynomial approximation algorithms (or even NC algorithms) for NP-complete graph problems when restricted to special classes of graphs. A wide class of such problems is defined in terms of hereditary properties (a graph property ß is called hereditary when, if ß is satisfied for some graph G, then ß is also satisfied for all induced subgraphs of G). The...

It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families. 1