The Number of Realizations of a Laman Graph
Copyright policy )The Number of Realizations of a Laman Graph
Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursive formula for the number of complex solutions of such systems.
36 pages
- University of Trieste Italy
graph embedding, Computer Science - Symbolic Computation, Computational Geometry (cs.CG), FOS: Computer and information sciences, Laman graph, minimally rigid graph, 14T05, 14N99, 52C25, 05C99, graph realization, Symbolic Computation (cs.SC), Projective and enumerative algebraic geometry, Combinatorial aspects of tropical varieties, Graph theory, Mathematics - Algebraic Geometry, tropical geometry, Rigidity and flexibility of structures (aspects of discrete geometry), Puiseux series, FOS: Mathematics, Euclidean embedding, Computer Science - Computational Geometry, Mathematics - Combinatorics, Combinatorics (math.CO), Algebraic Geometry (math.AG)
graph embedding, Computer Science - Symbolic Computation, Computational Geometry (cs.CG), FOS: Computer and information sciences, Laman graph, minimally rigid graph, 14T05, 14N99, 52C25, 05C99, graph realization, Symbolic Computation (cs.SC), Projective and enumerative algebraic geometry, Combinatorial aspects of tropical varieties, Graph theory, Mathematics - Algebraic Geometry, tropical geometry, Rigidity and flexibility of structures (aspects of discrete geometry), Puiseux series, FOS: Mathematics, Euclidean embedding, Computer Science - Computational Geometry, Mathematics - Combinatorics, Combinatorics (math.CO), Algebraic Geometry (math.AG)
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