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# Extending Nearly Complete 1-Planar Drawings in Polynomial Time

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LIPIcs.MFCS.2020.31.pdf
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## Cite As

Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending Nearly Complete 1-Planar Drawings in Polynomial Time. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.31

## Abstract

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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Extension problems
• 1-planarity

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