A Laman graph is a finite simple graph 
on 
graph vertices satisfying Laman's
theorem. It has exactly 
graph edges and satisfies 
for every subgraph
of 
having 
graph vertices and
graph edges.
The singleton graph 
is generally taken to be a Laman graph by convention, even
though it does not satisfy the condition 
.
Equivalently, apart from the singleton convention, Laman graphs are exactly the 
-tight case of the sparse-graph pebble game.
Laman graphs are always connected.
The numbers of Laman graphs on 
, 2, ... graph vertices are
1, 1, 1, 1, 3, 13, 70, 608, 7222, 110132, 2039273, 44176717, 1092493042, ... (OEIS
A227117).
Apart from the singleton convention, Laman graphs are minimally rigid in generic planar placements. Removing any graph edge makes such a placement flexible. Laman graphs form the bases of the two-dimensional rigidity matroids, which describe rigidity constraints on undirected graphs with graph edges of fixed lengths.
Nongeneric placements may have additional infinitesimal motions. For example, a planar placement of the utility graph 
has a rigidity matrix
of rank 9 iff its six graph
vertices do not lie on a conic. When they lie
on a conic, the placement admits an infinitesimal
motion other than a rigid motion (Bolker and Roth
1980, Maehara 1992).
Flexible embeddings of Laman graphs can commonly arise when incrementally removing sets of graph vertices from an initially rigid graph having a fair degree of symmetry.
The associated Laman number counts the number of generic complex planar realizations of a Laman graph up to direct complex isometry.
