Can words in regular languages be generated by automata in a unique way? If $L\subset \Sigma^\ast$ is a regular language then it is known that we can find a directed graph (aka automaton) $G$ with edges labelled with symbols from $\Sigma$, an ‘initial’ vertex, and a set of ‘terminal’ vertices, with the property that a word $w$ is in $L$ iff $w$ is the sequence of edge labels along a path from the initial vertex to a terminal vertex.

It’s possible that the same word can appear as a result of different paths in $G$. Can we guarantee that for any regular $L$ we can find a $G$ such that each word in $L$ appears as a path in $G$ in a unique way?

(I’m interested in this because I want to know whether the solution to the problem I worked on here applies cleanly to any regular language, or just for a certain class of them.)
Maybe I misunderstood your question, but wouldn't a Deterministic Finite Automaton do what you want?