3 research outputs found
Recognizing Planar Laman Graphs
Laman graphs are the minimally rigid graphs in the plane. We present two algorithms for recognizing planar Laman graphs. A simple algorithm with running time O(n^(3/2)) and a more complicated algorithm with running time O(n log^3 n) based on involved planar network flow algorithms. Both improve upon the previously fastest algorithm for general graphs by Gabow and Westermann [Algorithmica, 7(5-6):465 - 497, 1992] with running time O(n sqrt{n log n}).
To solve this problem we introduce two algorithms (with the running times stated above) that check whether for a directed planar graph G, disjoint sets S, T subseteq V(G), and a fixed k the following connectivity condition holds: for each vertex s in S there are k directed paths from s to T pairwise having only vertex s in common. This variant of connectivity seems interesting on its own
Combinatorial and Geometric Properties of Planar Laman Graphs
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.Comment: 17 pages, 11 figures, SODA 201