6,088 research outputs found
Primer for the algebraic geometry of sandpiles
The Abelian Sandpile Model (ASM) is a game played on a graph realizing the
dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of
this primer is to apply the theory of lattice ideals from algebraic geometry to
the Laplacian matrix, drawing out connections with the ASM. An extended summary
of the ASM and of the required algebraic geometry is provided. New results
include a characterization of graphs whose Laplacian lattice ideals are
complete intersection ideals; a new construction of arithmetically Gorenstein
ideals; a generalization to directed multigraphs of a duality theorem between
elements of the sandpile group of a graph and the graph's superstable
configurations (parking functions); and a characterization of the top Betti
number of the minimal free resolution of the Laplacian lattice ideal as the
number of elements of the sandpile group of least degree. A characterization of
all the Betti numbers is conjectured.Comment: 45 pages, 14 figures. v2: corrected typo
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
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