A Laman graph is a graph satisfying Laman's theorem. In other words, it is a graph 
have exactly 
graph edges, where 
is the number of graph vertices
in 
and for which 
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 
.
Laman graphs are always connected.
The numbers of simple Laman graphs on 
, 2, ... nodes are 1, 1, 1, 1, 3, 13, 70, 608, 7222, 110132,
2039273, 44176717, 1092493042, ... (OEIS A227117).
Laman graphs are minimally rigid, in the sense that removing any edge makes the resulting graph flexible when placed "in general position." Laman graphs form the bases of the so-called two-dimensional rigidity matroids, which are describe the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths.
Laman graphs are "generically" rigid in 
. In other words, they are rigid for
embeddings with vertices in general position.
However, a Laman graph may have flexible embeddings corresponding to vertices satisfying
constraints of lower dimension. For example, an embedding of the utility
graph 
,
which is a Laman graph, is rigid in the plane unless its six vertices lie on a conic (Bolker and Roth 1980, Maehara 1992).
Flexible embeddings of Laman graphs can commonly arise when incrementally removing sets of vertices from an initially rigid graph having a fair degree of symmetry.
