Combinatorial and Geometric Properties of Planar Laman Graphs

TL;DR

This paper introduces new combinatorial structures for planar Laman graphs, proves their equivalence, and demonstrates that these graphs can be represented as contact graphs of axis-aligned L-shapes with efficient algorithms.

Contribution

It presents three novel combinatorial structures for planar Laman graphs and shows their equivalence, enabling efficient algorithms for L-contact representations.

Findings

Planar Laman graphs have an angular structure called an angular tree.

Every angular tree corresponds to an angle and edge labeling.

All planar Laman graphs can be represented as contact graphs of axis-aligned L-shapes.

Abstract

Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling. Using a combination of these powerful combinatorial structures, we show that every planar Laman graph has an L-contact representation, that is, planar Laman graphs are contact graphs of axis-aligned L-shapes. Moreover, we show that planar Laman graphs and their subgraphs are the only graphs that can be represented this way. We present efficient algorithms that compute, for every planar Laman graph G, an angular tree, angle labeling, edge labeling, and finally an L-contact representation of G. The overall running time is O(n^2), where n is the number of vertices of G, and the L-contact representation is realized on the n x n grid.