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E. Eiben, R. Ganian, T. Hamm, F. Klute, M. Nöllenburg:
"Extending Nearly Complete 1-Planar Drawings in Polynomial Time";
Talk: International Symposium on Mathematical Foundations of Computer Science (MFCS), Prag, Tschechien; 2020-08-24 - 2020-08-28; in: "45th International Symposium on Mathematical Foundations of Computer Science", LIPIcs, 170 (2020), ISBN: 978-3-95977-159-7; 1 - 16.

The problem of extending partial geometric graph representations such as plane graphs has received
considerable attention in recent years. In particular, given a graph G, a connected subgraph H of G
and a drawing H of H, the extension problem asks whether H can be extended into a drawing of G
while maintaining some desired property of the drawing (e.g., planarity).
In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the extension
problem is polynomial-time solvable when the aim is to preserve planarity. Very recently we
considered this problem for partial 1-planar drawings [ICALP 2020], which are drawings in the plane
that allow each edge to have at most one crossing. The most important question identified and left
open in that work is whether the problem can be solved in polynomial time when H can be obtained
from G by deleting a bounded number of vertices and edges. In this work, we answer this question
positively by providing a constructive polynomial-time decision algorithm.

http://dx.doi.org/10.4230/LIPIcs.MFCS.2020.31

https://publik.tuwien.ac.at/files/publik_292473.pdf