A square-contact representation of a planar graph G = (V,E) maps vertices in V to interior-disjoint axis-aligned squares in the plane and edges in E to adjacencies between the sides of the corresponding squares. In this paper, we study proper square-contact representations of planar graphs, in which any two squares are either disjoint or share infinitely many points. We characterize the partial 2-trees and the triconnected cycle-trees allowing for such representations. For partial 2-trees our characterization uses a simple forbidden subgraph whose structure forces a separating triangle in any embedding. For the triconnected cycle-trees, a subclass of the triconnected simply-nested graphs, we use a new structural decomposition for the graphs in this family, which may be of independent interest. Finally, we study square-contact representations of general triconnected simply-nested graphs with respect to their outerplanarity index.
@InProceedings{dalozzo_et_al:LIPIcs.ISAAC.2017.24, author = {Da Lozzo, Giordano and Devanny, William E. and Eppstein, David and Johnson, Timothy}, title = {{Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.24}, URN = {urn:nbn:de:0030-drops-82675}, doi = {10.4230/LIPIcs.ISAAC.2017.24}, annote = {Keywords: Square-Contact Representations, Partial 2-Trees, Simply-Nested Graphs} }
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