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R. Ganian, T. Hamm, F. Klute, I. Parada, B. Vogtenhuber:
"Crossing-Optimal Extension of Simple Drawings";
Talk: International Colloquium on Automata, Languages and Programming (ICALP), Glasgow, Scotland; 2021-07-12 - 2021-07-16; in: "48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)", LIPICS, 198 (2021), ISBN: 978-3-95977-195-5; 1 - 17.

In extension problems of partial graph drawings one is given an incomplete drawing of an input graph
G and is asked to complete the drawing while maintaining certain properties. A prominent area where
such problems arise is that of crossing minimization. For plane drawings and various relaxations of
these, there is a number of tractability as well as lower-bound results exploring the computational
complexity of crossing-sensitive drawing extension problems. In contrast, comparatively few results
are known on extension problems for the fundamental and broad class of simple drawings, that is,
drawings in which each pair of edges intersects in at most one point. In fact, the extension problem
of simple drawings has only recently been shown to be NP-hard even for inserting a single edge.
In this paper we present tractability results for the crossing-sensitive extension problem of simple
drawings. In particular, we show that the problem of inserting edges into a simple drawing is
fixed-parameter tractable when parameterized by the number of edges to insert and an upper bound
on newly created crossings. Using the same proof techniques, we are also able to answer several
closely related variants of this problem, among others the extension problem for k-plane drawings.
Moreover, using a different approach, we provide a single-exponential fixed-parameter algorithm for
the case in which we are only trying to insert a single edge into the drawing.

http://dx.doi.org/10.4230/LIPIcs.ICALP.2021.72

https://publik.tuwien.ac.at/files/publik_300282.pdf