For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes G^{hom}(A) and G^{sim}(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that G^{hom}(A) = G^{hom}(B) if and only if A and B are affine equivalent, and G^{sim}(A) = G^{sim}(B) if and only if A and B are similar.
@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2023.2, author = {Abrahamsen, Mikkel and Walczak, Bartosz}, title = {{Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {2:1--2:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.2}, URN = {urn:nbn:de:0030-drops-178523}, doi = {10.4230/LIPIcs.SoCG.2023.2}, annote = {Keywords: geometric intersection graph, convex disk, homothet, similarity} }
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