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 manuscript focusses on algebraic shifting and its applications to f-vector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the g-conjecture for simplicial spheres, which is considered by many as the main open problem in f-vector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in f-vector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds.

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

Gluck [6] has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K5-minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2-minor free graph is generically r-stress free for 1 ≤ r ≤ 4. (This assertion is false for r ≥ 6.) Some further extensions are discussed. 1 Introduction and

This manuscript focusses on algebraic shifting and its applications to f-vector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the g-conjecture for simplicial spheres, which is considered by many as the main open problem in f-vector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in f-vector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds.