Axis-Aligned Square Contact Representations

TL;DR

This paper introduces a new class of bipartite plane graphs and proves that each admits a proper square contact representation, with a focus on the aspect ratio variability of the outer rectangle in the construction.

Contribution

The paper defines the class $ ext{G}$ of quadrangulations and proves that all graphs in this class have proper square contact representations with variable aspect ratios.

Findings

Proper square contact representations exist for all graphs in class $ ext{G}$.

The aspect ratio of the outer rectangle can vary continuously within an interval.

The feasible aspect ratio interval can be arbitrarily small, near any positive real number.

Abstract

We introduce a new class $\mathcal{G}$ of bipartite plane graphs and prove that each graph in $\mathcal{G}$ admits a proper square contact representation. A contact between two squares is \emph{proper} if they intersect in a line segment of positive length. The class $\mathcal{G}$ is the family of quadrangulations obtained from the 4-cycle $C_4$ by successively inserting a single vertex or a 4-cycle of vertices into a face. For every graph $G\in \mathcal{G}$, we construct a proper square contact representation. The key parameter of the recursive construction is the aspect ratio of the rectangle bounded by the four outer squares. We show that this aspect ratio may continuously vary in an interval $I_G$. The interval $I_G$ cannot be replaced by a fixed aspect ratio, however, as we show, the feasible interval $I_G$ may be an arbitrarily small neighborhood of any positive real.