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A273468
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Number of minimally rigid graphs with n vertices constructible by Henneberg type I moves.
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5
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1, 1, 1, 1, 3, 11, 61, 499, 5500, 75635, 1237670, 23352425, 498028767, 11836515526, 310152665647
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OFFSET
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1,5
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COMMENTS
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A graph is called rigid if, when we fix the length of each edge, it has only finitely many embeddings in the plane. A graph is called minimally rigid (or a Laman graph) if there is no edge that can be omitted while keeping the rigidity property. Laman graphs can be constructed by applying successively Henneberg moves (of type I or type II), starting with the graph that consists of two vertices joined by an edge. Here we consider Laman graphs that can be obtained by using only Henneberg type I moves, which means: adding one vertex and joining it with two different existing vertices.
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LINKS
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Table of n, a(n) for n=1..15.
L. Henneberg, Die graphische Statik der starren Systeme, Leipzig, 1911.
Christoph Koutschan, Mathematica program
G. Laman, On Graphs and Rigidity of Plane Skeletal Structures, Journal of Engineering Mathematics 4 (1970), 331-340.
Martin Larsson, C program
Wikipedia, Laman graph
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EXAMPLE
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A single vertex is rigid.
The graph consisting of two vertices joined by an edge is rigid.
A triangle is rigid and it is obtained by a single Henneberg type I move.
One more such move yields the only Laman graph with four vertices.
Also all three Laman graphs with five vertices can be constructed with type I moves. Therefore the first five entries of this sequence agree with A227117.
An example of a Laman graph that cannot be constructed using only Henneberg type I moves is the complete bipartite graph K(3,3).
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MATHEMATICA
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Table[Length[H1LamanGraphs[n]], {n, 3, 7}] (* see link *)
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CROSSREFS
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Cf. A227117.
Sequence in context: A095237 A185385 A024528 * A004108 A203007 A343774
Adjacent sequences: A273465 A273466 A273467 * A273469 A273470 A273471
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KEYWORD
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nonn,more
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AUTHOR
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Christoph Koutschan, May 23 2016
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EXTENSIONS
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a(13) added by Jose Capco, Dec 07 2018
a(14)-a(15) added by Martin Larsson, Dec 21 2020
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STATUS
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approved
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