doi:10.26493/1855-3974.2226.E93engGarcía Olaverri A.Tejel Altarriba J.Pilz A.On plane subgraphs of complete topological drawingsART-2021-127326Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a common endpoint or at a proper crossing. In this paper we study properties of maximal plane subgraphs of simple drawings Dnof the complete graph Knon n vertices. Our main structural result is that maximal plane subgraphs are 2-connected and what we call essentially 3-edge-connected. Besides, any maximal plane subgraph contains at least [3n/2] edges. We also address the problem of obtaining a plane subgraph of Dnwith the maximum number of edges, proving that this problem is NP-complete. However, given a plane spanning connected subgraph of Dn, a maximum plane augmentation of this subgraph can be found in O(n3) time. As a side result, we also show that the problem of finding a largest compatible plane straight-line graph of two labeled point sets is NP-complete. © 2021 Society of Mathematicians, Physicists and Astronomers of Slovenia. All rights reserved.2021http://zaguan.unizar.es/record/11059910.26493/1855-3974.2226.E93http://zaguan.unizar.es/record/110599oai:zaguan.unizar.es:110599info:eu-repo/grantAgreement/ES/DGA/E41-17Rinfo:eu-repo/grantAgreement/EC/H2020/734922/EU/Combinatorics of Networks and Computation/CONNECTThis project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No H2020 734922-CONNECTinfo:eu-repo/grantAgreement/ES/MICIU-AEI/PID2019-104129GB-I00-AEI-10.13039-501100011033info:eu-repo/grantAgreement/ES/MINECO/MTM2015-63791-RArs Mathematica Contemporanea 20, 1 (2021), 69-87byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess