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# Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs

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## Cite As

Giordano Da Lozzo, William E. Devanny, David Eppstein, and Timothy Johnson. Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ISAAC.2017.24

## Abstract

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.
##### Keywords
• Square-Contact Representations
• Partial 2-Trees
• Simply-Nested Graphs

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## References

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