Just to quickly type down some notes for myself and to establish a semi-serious starting point of effort on this problem.
===
0.0. The object we consider belongs to a special kind of graphs which we shall define later. Here, whenever speaking of graphs, we refer to simple, undirected, connected graphs.
0.1. A graph is said to triangulated if any cycle of size no less than 4 has a chord.
0.2. A graph may be embedded in an Euclidean space, namely, having each of its vertices associated with a coordinate in a Euclidean space. In particular, the Euclidean space we consider is going to be R^2, the two dimensional vector space over the reals. In this case, a graph becomes a geometric graph.
0.3. We are interested in a particular kinds of geometric graphs, called planar straight-line graphs, in which each edge in the graph is a straight line segment in R^2. From here on, whenever speaking of graphs, we exclusively refer to planar straight-line graphs in which each edge has length precisely 1.
0.4. Further without loss of generality, we will assume that in a graph, there always exists a vertex whose coordinate is (0,0), the origin.
0.5. Two graphs are said to be equivalent, if it is possible to rotate/translate/reflect one graph to obtain the other.
Now I am trying to define a notion of "stability", which I call "rigid".
Definition 1. [Rigid] A graph G is said to be rigid if any other graph H that is graph-theoretically isomorphic to G is equivalent to G.
(to be continued ...)