Natural realizations of sparsity matroids

A hypergraph G with n vertices and m hyperedges with d endpoints each is (k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a linearly representable matroidal family. Motivated by problems in rigidity theory, we give a new linear representation theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the representing matrix captures the vertex-edge incidence structure of the underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars Mathematica Contemporane

Slider-pinning Rigidity: a Maxwell-Laman-type Theorem

We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley's direction networks.Comment: Accepted, to appear in Discrete and Computational Geometr

Sparsity-Certifying Graph Decompositions

We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]

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

Enumerating grid layouts of graphs

We study algorithms that generate layouts of graphs with n vertices in a square grid with ν points, where adjacent vertices in the graph are also close in the grid. The problem is motivated by graph drawing and factory layout planning. In the latter application, vertices represent machines, and edges join machines that should be placed next to each other. Graphs admitting a grid layout where all edges have unit length are known as partial grid graphs. Their recognition is NP-hard already in very restricted cases. However, the moderate number of machines in practical instances suggests the use of exact algorithms that may even enumerate the possible layouts to choose from. We start with an elementary nO(√n)\ua0time algorithm, but then we argue that even simpler exponential branching algorithms are more usable for practical sizes n, although being asymptotically worse. One algorithm interpolates between obvious O∗(3n) time and O∗(4ν) time for graphs with many small connected components. It can be modified in order to accommodate also a limited number of edges that can exceed unit length. Next we show that connected graphs have at most 2.9241n\ua0grid layouts that can also be efficiently enumerated. An O∗(2.6458n) time branching algorithm solves the recognition problem, or yields a succinct enumeration of layouts with some surcharge on the time bound. In terms of the grid size we get a slightly better O∗(2.6208ν) time bound. Moreover, if we can identify a subgraph that is rigid, i.e., admits only one layout up to congruence, then all possible layouts of the entire graph are extensions of this unique layout, such that the combinatorial explosion is then confined to the rest of the graph. Therefore we also propose heuristic methods for finding certain types of large rigid subgraphs. The formulations of these results is more technical, however, the proposed method iteratively generates certain rigid subgraphs from smaller ones