An independent set in a graph is a set of vertices where no two vertices are adjacent to each other. A maximum independent set is the largest possible independent set that can be formed within a given graph G. The cardinality of this set is referred to as the independence number of G. This paper investigates the independence number of 1-planar graphs, a subclass of graphs defined by drawings in the Euclidean plane where each edge can have at most one crossing point. Borodin establishes a tight upper bound of six for the chromatic number of every 1-planar graph G, leading to a corresponding lower bound of n/6 for the independence number, where n is the number of vertices of G. In contrast, the upper bound for the independence number in 1-planar graphs is less studied. This paper addresses this gap by presenting upper bounds based on the minimum degree δ. A comprehensive table summarizes these upper bounds for various δ values, providing insights into achievable independence numbers under different conditions.
@InProceedings{biedl_et_al:LIPIcs.SWAT.2024.13, author = {Biedl, Therese and Bose, Prosenjit and Miraftab, Babak}, title = {{On the Independence Number of 1-Planar Graphs}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.13}, URN = {urn:nbn:de:0030-drops-200537}, doi = {10.4230/LIPIcs.SWAT.2024.13}, annote = {Keywords: 1-planar graph, independent set, minimum degree} }
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