Traditional problems in graph visualization involve the layout of a single graph, while problems in simultaneous graph visualization involve the layout of multiple related graphs. A series of related graphs may arise from one relation in a set of objects as it evolves through time, or from several relationships defined on the same set of objects. In simultaneous embedding, nodes are placed in the exact same locations in all the graphs and a series of graphs is simultaneously embedable if it is possible to find node locations that yield straight-line crossing-free drawings for each of the individual graphs.

We first present set of unlabeled level planar (ULP) graphs that are level planar over all possible labelings.  Our contributions are twofold: 1) we provide linear time drawing algorithms for ULP graphs; 2) we provide a complete characterization of ULP graphs. ULP graphs also turn out to play a role in problems related to level planarity.

In the second part of the talk we present minimum level nonplanar patterns (MLNP) patterns for trees. MLNP patterns play the role for level planar trees that the forbidden Kuratowski subdivisions $K_5$ and $K_{3,3}$ play for planar graphs.  We add two MLNP patterns for trees to the previous set of tree patterns given by Healy et al.  Neither of these patterns match any of the previous patterns and thus our results correct an error that has gone unnoticed in five years.  We show that this new set of patterns completely characterize level planar trees.