Axis-Aligned Square Contact Representations

We introduce a new class $\mathcal{G}$ of plane bipartite 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.