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arXiv · 1710.00426

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

Giordano Da Lozzo·William E. Devanny·David Eppstein·Timothy Johnson
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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.

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Giordano Da Lozzo, William E. Devanny, David Eppstein, Timothy Johnson. 2017-10-01. Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs. https://arxiv.org/abs/1710.00426

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