OFFSET
1,3
COMMENTS
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.
LINKS
Eric A. Butcher and Chris Hartman, Efficient enumeration and hierarchical classification of planar simple-jointed kinematic chains: Application to 12- and 14-bar single degree-of-freedom chains, Mechanism and Machine Theory 30 (2005), 1030-1050.
Audrey Lee and Ileana Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (2008), 1425-1437.
E. E. Peisakh, Structural analysis of planar jointed mechanisms: Current state and problems, J. Machinery Manufacture and Reliability 37 (2008), 207-212.
Rajesh P. Sunkari and Linda C. Schmidt, Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm, Mechanism and Machine Theory 41 (2006), 1021-1030. This paper sources the 19819281 value for n=8 but contains a typo for n=7.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
David Eppstein, Dec 06 2013
STATUS
approved