# Is there a way to represent edges in graph theory that may only be used once?

I've been working on taking sentences, converting morphemes into nodes and then having edges connecting the nodes in the different possible syntactically correct orders of a sentence. However, in doing this, the graphs that I create sometimes look like they allow for infinite cycles whereas in fact they do not. For example, I have this graph

which would seem to imply that there is a cycle $\sigma(V,I,D)$, which does not exist. Is there a way to indicate that only certain paths are possible and that no cycles exist? What I'm trying to show is that $S$ must be the origination and from $S$ you can go to $V$, $I$, or $D$. Then from each of those you can go to the other two in certain orders. The paths I'm trying to graph are as such \begin{align} S \to I \to D \to V \\ S \to I \to V \to D \\ S \to D \to V \to I \\ S \to V \to D \to I \end{align} From the included graph all of those paths can be followed, but other non-existent paths can also be followed. Is there a better way to notate these specific paths within the above graph or is there a different methodology that should be used?

• Presumably the paths are always of length 4. And just to clarify, SVID and SDIV are not allowed? – bvy Dec 4 '16 at 18:36
• @bvy The paths are always of length 4. Yes $S\to V\to I\to D$ and $S\to D\to I\to V$ are not allowed. – Eli Sadoff Dec 4 '16 at 18:43
• The rules suggest a tree, but you probably don't want multiple vertices representing a single morpheme. I don't see a way to model this with a simple digraph. Perhaps some sort of flow network, but it's not my area of expertise. – bvy Dec 4 '16 at 19:24
• Ok but why is $S \to V \to I \to D$ (for example) not allowed? What is the rule? Because as I see it, you are allowed to move from $V \to I$, but only under special circumstances. (agree with bvy that it seems simple digraphs are not the best way to represent) – Morgan Rodgers Dec 4 '16 at 19:24
• @MorganRodgers I'm mapping a french grammatical structure. So you can't for example say "J'ai donné à toi ça" because that doesn't follow grammatical rules whereas "J'ai donné ça à toi" does. – Eli Sadoff Dec 4 '16 at 19:30