How many dihedral angles need to be specified to uniquely specify a triangulated polyhedron? Suppose you are given a simplicial complex $K$ homeomorphic to the sphere and for each each edge of the complex a label specifying a length of that edge (this gives us a polyhedral metric on $K$). In other words, I'm specifying the local geometry and combinatorics of a polyhedron, but not its realization.
We'll call a realization of $K$ a polyhedron in $\mathbb{R}^3$ whose vertex, edge, and face structure is given by $K$ and whose edges are equivalent to the given edge length labeling. Suppose also that I know that at least one realization exists (side question: is it NP-hard to determine this?).
I know that thanks to the familiar examples, like the "house" made of a cube (with triangulated sides for consistency) and a pyramid on top (and its alternative the house with an inverted roof), the edge labeling is not enough to uniquely specify the realization (barring something like requiring convexity and then invoking Cauchy's theorem). On the other hand, clearly if I specify all the dihedral angles as well I've uniquely specified the realization, but this feels like overkill. My question is: beyond knowing the edge lengths, how much extra information is needed to uniquely specify the polyhedron? It seems pretty clear that if I take a spanning tree of the dual graph, and specify the dihedral angles represented by each dual edge then that should be enough, but even this feels like it's too much information. Can I always get away with less? 
Any references to work on this or related problem is appreciated. 
 A: First, you didn't say this explicitly in the question, but I assume you only care about realizations up to isometries of 3D Euclidean space. (Another possible choice would be realizations up to rigid body motions (orientation-preserving isometries), but then mirror-image realizations can't be distinguished by any set of dihedral angles.)
I don't have a complete answer, but I can at least give some examples showing that the minimal amount of extra information required will depend on the combinatorics. First, it's possible that one dihedral angle is enough to distinguish all realizations (let's leave aside triangulations with unique realizations which could be said to require 0 dihedrals). Consider for instance a realizable triangular bipyramid with generic edge lengths (figure from Fedorchuk and Pak, linked below):

There are generically two realizations, one convex and one non-convex, and these can be distinguished by the dihedral angle at any of the three edges which connects two vertices of degree 4.
More generally, take two triangulations $T_1$ and $T_2$ with edge labelings which agree on a triangle $t$, and let's assume that this leads to unique realizations (I don't a general construction, but let me assume that many of these exist; my guess is that "most" convex 4-connected triangulations have a unique realization). Then we can get a new triangulation $T'$ with two realizations by gluing $T_1$ and $T_2$ along $t$. Just as above, these two realizations can be distinguished by considering just one dihedral angle.
Note that the triangular bipyramid can be constructed by replacing a triangular face of a tetrahedron by three triangles connected to a degree 3 vertex. Let's call this operation a "Henneberg-I move" (this is terminology from rigidity theory, see e.g. this survey).
It's not hard to see that each time you perform a Henneberg-I move on a triangulation, you double the number of realizations, and the dihedral angles of the pairs of realizations are equal except on the edges of the removed triangle. Thus by performing a sequence of Henneberg-I moves on a tetrahedron you can construct triangulations of the sphere whose realizations can only be distinguished by considering many dihedral angles at once. In particular, this leads to triangulations with $2^m$ realizations which require $m$ dihedral angles to be distinguished. 

Are there families of triangulations which require asymptotically more dihedrals (per realization, say) to distinguish their realizations?

As for related literature, the following might be interesting:


*

*"Rigidity and polynomial invariants of convex polytopes" by Fedorchuk and Pak) (check out also Pak's lecture notes for lots more information about rigidity of polytopes). I think the (cosines of the) dihedrals are polynomial invariants in the sense of this paper. 

*Jürgen Richter-Gebert also has a book titled "Realization Spaces of Polytopes" with lots of valuable information. For instance, Theorem 13.3.3 shows that the realization space of a 3-polytope with $e$ edges is homeomorphic to an open ball of dimension $e-6$. (Note that in this definition, affine transformations are factored out).

*There are a number of papers on the problem of counting realizations of graphs given edge lengths, here is an early one by Borcea and Streinu and here is a more recent one by Bartzos, Emiris, Legerský, and Tsigaridas.

*Regarding the side question of hardness of determining whether a realization exists, here are some related papers. Saxe showed that the problem of determining existence of $k$-dimensional realizations of arbitrary graphs in $\mathbb{R}^k$ with $k\geq 2$, is NP-hard. Here is a paper by Cabello, Demaine and Rote on the hardness of finding planar realizations of planar graphs. However, I do not know what happens for triangulations in $\mathbb{R}^3$ with generic edge labelings and would be very curious to know the answer as well.
A: It is an open problem to determine if the combinatorics plus the dihedrals
determine a convex polyhedron.
More precisely Stoker's conjecture from 1968—still unsolved—says that the face lattice of a convex polyhedron and its dihedral angles determine the face angles:

Stoker, James J. "Geometrical problems concerning polyhedra in the large." Communications on Pure and Applied Mathematics 21, no. 2 (1968): 119-168.

This doesn't address the OP's "beyond knowing the edge lengths,...",
but rather what is needed beyond knowing the dihedral angles (for convex polyhedra).
There has been relatively recent progress on Stoker's conjecture:

Mazzeo, Rafe, and Grégoire Montcouquiol. "Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra." Journal of Differential Geometry 87, no. 3 (2011): 525-576.
  (arXiv abs.)
"We prove here an infinitesimal rigidity result valid for cone angles less than $2\pi$, stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case."
  
            
  


