Rigidity
Laman graphs arise in rigidity theory: if one places the vertices of a Laman graph in the Euclidean plane, in general position, there will in general be no simultaneous motion of all the points, other than Euclidean congruences, that preserves the lengths of all the graph edges. A graph is rigid in this sense if and only if it has a Laman subgraph that spans all of its vertices. Thus, the Laman graphs are exactly the minimally rigid graphs, and they form the bases of the two-dimensional rigidity matroids.
If n points in the plane are given, then there are 2n degrees of freedom in their placement (each point has two independent coordinates), but a rigid graph has only three degrees of freedom (the position of a single one of its vertices and the rotation of the remaining graph around that vertex). Intuitively, adding an edge of fixed length to a graph reduces its number of degrees of freedom by one, so the 2n − 3 edges in a Laman graph reduce the 2n degrees of freedom of the initial point placement to the three degrees of freedom of a rigid graph. However, not every graph with 2n − 3 edges is rigid; the condition in the definition of a Laman graph that no subgraph can have too many edges ensures that each edge contributes to reducing the overall number of degrees of freedom, and is not wasted within a subgraph that is already itself rigid due to its other edges.
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Famous quotes containing the word rigidity:
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