The digraph realization problem is a decision problem in graph theory. Given pairs of nonnegative integers
((a1,b1),\ldots,(an,bn))
vi
ai
bi
The problem belongs to the complexity class P. Two algorithms are known to prove that. The first approach is given by the Kleitman–Wang algorithms constructing a special solution with the use of a recursive algorithm. The second one is a characterization by the Fulkerson–Chen–Anstee theorem, i.e. one has to validate the correctness of
n
The problem can also be stated in terms of zero-one matrices. The connection can be seen if one realizes that each directed graph has an adjacency matrix where the column sums and row sums correspond to
(a1, … ,an)
(b1,\ldots,bn)
Similar problems describe the degree sequences of simple graphs, simple directed graphs with loops, and simple bipartite graphs. The first problem is the so-called graph realization problem. The second and third one are equivalent and are known as the bipartite realization problem. gives a characterization for directed multigraphs with a bounded number of parallel arcs and loops to a given degree sequence. The additional constraint of the acyclicity of the directed graph is known as dag realization. proved the NP-completeness of this problem. showed that the class of opposed sequences is in P. The problem uniform sampling a directed graph to a fixed degree sequence is to construct a solution for the digraph realization problem with the additional constraint that such each solution comes with the same probability. This problem was shown to be in FPTAS for regular sequences by The general problem is still unsolved.