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.

This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open

this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1

A new characterisation of planar graphs is presented. It concerns the structure of the cocycle space of a graph, and is motivated by consideration of the dual of an elementary property enjoyed by sets of circuits in any graph. 1