Graph realization problem

The graph realization problem is a decision problem in graph theory. Given a finite sequence ${\displaystyle (d_{1},\dots ,d_{n})}$ of natural numbers, the problem asks whether there is a labeled simple graph such that ${\displaystyle (d_{1},\dots ,d_{n})}$ is the degree sequence of this graph.

In the context of localization, the graph realization problem may also refer to finding a set of positions ${\displaystyle (x_{1},\dots ,x_{n})}$ in some Euclidean space such that the squared distances between the positions, given by ${\displaystyle d_{ij}^{2}}$, match the edge weights ${\displaystyle w_{ij}}$ for all edges in an incomplete, undirected, weighted graph.