Given a family of (hyper) graphs $\mathcal {F} $ a (hyper) graph $ G $ is said to be $\mathcal
{F} $-saturated if $ G $ is $ F $-free for any $ F\in\mathcal {F} $ but for any edge e in the …

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the
sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the …

A graph is $2 $-planar if it has local crossing number two, that is, it can be drawn in the
plane such that every edge has at most two crossings. A graph is maximal $2 $-planar if no …

A drawing of a graph is $ k $-plane if every edge contains at most $ k $ crossings. A $ k $-
plane drawing is saturated if we cannot add any edge so that the drawing remains $ k …

Before we attack state-of-the-art problems of graph coloring including but not limited to the
planar graphs in Chapter 3, we will set up the notation and defnitions in this thesis in …

This thesis revolves around the topic of beyond-planar graphs. Planar graphs are graphs
that can be drawn on the plane without any crossings. Beyond-planar graphs are a …

For a class D of drawings of loopless (multi-) graphs in the plane, a drawing D∈ D is
saturated when the addition of any edge to D results in D'∉ D-this is analogous to saturated …