of natural numbers, the problem asks whether there is a labeled simple graph such that
In the context of localization, the graph realization problem may also refer to finding a set of positions
in some Euclidean space such that the squared distances between the positions, given by
for all edges in an incomplete, undirected, weighted graph.
[1] The problem can be solved in polynomial time.
One method of showing this uses the Havel–Hakimi algorithm constructing a special solution with the use of a recursive algorithm.
[2][3] Alternatively, following the characterization given by the Erdős–Gallai theorem, the problem can be solved by testing the validity of
[4] The problem can also be stated in terms of symmetric matrices of zeros and ones.
The connection can be seen if one realizes that each graph has an adjacency matrix where the column sums and row sums correspond to
The problem is then sometimes denoted by symmetric 0-1-matrices for given row sums.
Similar problems describe the degree sequences of simple bipartite graphs or the degree sequences of simple directed graphs.
The problem of constructing a solution for the graph realization problem with the additional constraint that each such solution comes with the same probability was shown to have a polynomial-time approximation scheme for the degree sequences of regular graphs by Cooper, Martin, and Greenhill.