The problem is to realize or find a graph (i.e. say adjacency matrix) given one fundamental cut-set matrix. I know that this problem is not complete, since we run into the problem of 2-isomorphism, hence uniqueness cannot be guaranteed. 

Is there any method by which I can easily realize at least one possible/compatible graph?

This broadly falls under graph realization problem, which had some activity till the late 1980s but seems dead after that. I was also not able to find any comprehensive algorithm to do this task. If anyone can help me out, it would be great. Thanks!

The latest paper that I have come across is 1985 paper ' Almost Linear-time Algorithm for Graph Realization' by Robert Bixby