Rectangle and square representations of planar graphs

Abstract

In the first part of this survey, we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings, the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs, the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures. In the second part, we ask for representations by a dissections of a rectangle into squares. We review results by Brooks et al. [The dissection of rectangles into squares. Duke Math J 7:312-340 (1940)], Kenyon [Tilings and discrete Dirichlet problems. Isr J Math 105:61-84 (1998)], and Schramm [Square tilings with prescribed combinatorics. Isr J Math 84:97-118 (1993)] and discuss a technique of computing squarings via solutions of systems of linear equations.