Radial Level Planarity with Fixed Embedding Vol. 25, no. 1, pp. 353-366, 2021. Regular paper. Abstract We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions of combinatorial embeddings, namely the level embedding, the upward embedding and the planar embedding. These notions allow for increasing degrees of freedom in their corresponding drawings. For the fixed level embedding there are known and easy to test level planarity criteria. We use these criteria to prove an "untangling" lemma that plays a key role in a simple level planarity test for the case where only the upward embedding is fixed. This test is then adapted to the case where only the planar embedding is fixed. Further, we characterize radial upward planar embeddings, which lets us extend our results to radial level planarity. No algorithms were previously known for these problems.  This work is licensed under the terms of the CC-BY license. Submitted: November 2020. Reviewed: March 2021. Revised: March 2021. Accepted: April 2021. Final: April 2021. Published: May 2021. Communicated by Giuseppe Liotta article (PDF) BibTeX