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

TL;DR

This paper characterizes which partial 2-trees and triconnected simply-nested graphs admit proper square-contact representations, providing structural insights and decompositions for these classes of planar graphs.

Contribution

It introduces a forbidden subgraph characterization for partial 2-trees and a new structural decomposition for triconnected cycle-trees, advancing understanding of square-contact representations.

Findings

Characterized partial 2-trees allowing proper square-contact representations.

Developed a structural decomposition for triconnected cycle-trees.

Analyzed square-contact representations in relation to outerplanarity index.

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.