58 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 Resultants in the Algebraic Rigidity Matroid
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid CMn associated to the Cayley-Menger ideal for n points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from K4 graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs
An inductive construction of (2,1)-tight graphs
The graphs with that satisfy for any subgraph (and for ) are the -tight graphs. The Henneberg--Laman theorem characterizes -tight graphs inductively in terms of two simple moves, known as the Henneberg moves. Recently, this has been extended, via the addition of a graph extension move, to the case of -tight simple graphs. Here an alternative characterization is provided by means of vertex-to- and edge-to- moves. This is extended to the -tight simple graphs by the addition of an edge joining move
Combinatorial resultants in the algebraic rigidity matroid
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid CMn associated to the Cayley-Menger ideal for n points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from K4 graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs
Nucleation-free rigidity
When all non-edge distances of a graph realized in as a {\em
bar-and-joint framework} are generically {\em implied} by the bar (edge)
lengths, the graph is said to be {\em rigid} in . For ,
characterizing rigid graphs, determining implied non-edges and {\em dependent}
edge sets remains an elusive, long-standing open problem.
One obstacle is to determine when implied non-edges can exist without
non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal
with them.
In this paper, we give general inductive construction schemes and proof
techniques to generate {\em nucleation-free graphs} (i.e., graphs without any
nucleation) with implied non-edges. As a consequence, we obtain (a) dependent
graphs in that have no nucleation; and (b) nucleation-free {\em
rigidity circuits}, i.e., minimally dependent edge sets in . It
additionally follows that true rigidity is strictly stronger than a tractable
approximation to rigidity given by Sitharam and Zhou
\cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial
characterization.
As an independently interesting byproduct, we obtain a new inductive
construction for independent graphs in . Currently, very few such inductive
constructions are known, in contrast to
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