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A233288
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Number of (3/2,2)-tight graphs with 2n vertices, or kinematic chains with 2n links.
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0
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OFFSET
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1,3
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COMMENTS
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A 2n-vertex graph is (3/2,2)-sparse if every subgraph with k vertices has at most (3/2)k-2 edges, and (3/2,2)-tight if in addition it has exactly 3n-2 edges; see Lee and Streinu (2008). These graphs represent two-dimensional mechanical systems formed by 2n rigid bodies (links), connected at joints where exactly two links are pinned together and can rotate relative to each other, with the entire system having one degree of freedom and having no rigid subsystems. The vertices of the graph represent links and the edges represent joints.
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LINKS
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EXAMPLE
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For n=1 the single example (a graph with two vertices and one edge) is represented by familiar mechanical systems including door hinges and pairs of scissors. For n=3 the a(3)=2 solutions are the 6-vertex 7-edge graphs Theta(1,3,3) and Theta(2,2,3), each of which has two degree-three vertices connected by three paths of the given lengths. These correspond respectively to the Watt linkage (two four-bar linkages sharing a pair of adjacent links) and the Stephenson linkage.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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