Many times when working with graphs, we need a graph representation from a given degree sequence. Whether that is for generating a sample network for world models or to find the different structural isomers of the same molecular formula.
For a given sequence of positive integers, we would like to know if a graph with that degree sequence exists, if so we want to construct a realization. But in the real world, we are not always interested in any such graph realization. Many times we require different restrictions on the graph. We thus limit the problem to connected graphs, trees, bipartite graphs and directed graphs. With an exhaustive method we then tackle the enumeration problem. We present our results for the enumeration of all realizations for a given degree sequence, with the different graph restrictions in mind.
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