A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings

TL;DR

This paper demonstrates that all planar graphs with a rectangular dual can be represented with plus-contact representations that are balanced with c=1/2, leading to efficient box-orthogonal drawings with controlled box sizes.

Contribution

It introduces a method to compute (1/2)-balanced plus-contact representations for all planar graphs with a rectangular dual, enabling improved box-orthogonal drawings.

Findings

Existence of (1/2)-balanced plus-contact representations for all such graphs.

Implication for box-orthogonal drawings with square boxes of size proportional to vertex degree.

Enhanced understanding of geometric representations of planar graphs.

Abstract

A plus-contact representation of a planar graph $G$ is called $c$-balanced if for every plus shape $+_v$, the number of other plus shapes incident to each arm of $+_v$ is at most $ c \Delta +O(1)$, where $\Delta$ is the maximum degree of $G$. Although small values of $c$ have been achieved for a few subclasses of planar graphs (e.g., $2$- and $3$-trees), it is unknown whether $c$-balanced representations with $c<1$ exist for arbitrary planar graphs. In this paper we compute $(1/2)$-balanced plus-contact representations for all planar graphs that admit a rectangular dual. Our result implies that any graph with a rectangular dual has a 1-bend box-orthogonal drawings such that for each vertex $v$, the box representing $v$ is a square of side length $\frac{deg(v)}{2}+ O(1)$.