A simultaneous representation of (vertex-labeled) graphs G_1,… ,G_k consists of a (geometric) intersection representation R_i for each graph G_i such that each vertex v is represented by the same geometric object in each R_i for which G_i contains v. While Jampani and Lubiw showed that the existence of simultaneous interval representations for k = 2 can be tested efficiently (2010), testing it for graphs where k is part of the input is NP-complete (Bok and Jedličková, 2018). An important special case of simultaneous representations is the sunflower case, where G_i ∩ G_j = (V(G_i)∩ V(G_j),E(G_i)∩ E(G_j)) is the same graph for each i ≠ j. We give an O(∑_{i=1}^k (|V(G_i)|+|E(G_i)|))-time algorithm for deciding the existence of a simultaneous interval representation for the sunflower case, even when k is part of the input. This answers an open question of Jampani and Lubiw.
@InProceedings{rutter_et_al:LIPIcs.ESA.2023.90, author = {Rutter, Ignaz and Stumpf, Peter}, title = {{Simultaneous Representation of Interval Graphs in the Sunflower Case}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {90:1--90:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.90}, URN = {urn:nbn:de:0030-drops-187435}, doi = {10.4230/LIPIcs.ESA.2023.90}, annote = {Keywords: Interval Graphs, Sunflower Case, Simultaneous Representation, Recognition, Geometric Intersection Graphs} }
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