I am not sure I understand the question correctly. If not let me know.

This simple method might not be optimal but it will work:

1 2 3 4* 2 3 4* 4 3*\EOL

This indicates the the following directed outgoing edges

from 1 to 2, 3, 4,

from 2 to 3 and 4,

from 4 to 3

and there are no more edges.

If you are particular about efficiency then I am sure that someone might readily know some references.

Best.

There are a few algorithms out there for chemical applications that convert molecular graphs into line representations. One output format is called the SMILES format. http://pubs.acs.org/doi/abs/10.1021/ci00062a008

you could use the same concept to convert your graph into a single word. If the number of of nodes in your graph is n, then the number of characters in your "SMILES" word is also n

Thank you much!