# Is there a term for 'no-turn-back walk' in graph theory?

Let $$G$$ be a finite undirected graph. A walk in $$G$$ is a finite sequence $$$$ where $$v_j$$'s are vertices in $$G$$, $$e_j$$'s are edges in $$G$$, and $$e_j=v_jv_{j+1}$$ for $$j. Suppose we add a condition that no edge is visited again in the next step itself; that is, $$e_{j+1}\neq e_j$$ for $$1\leq j< k-2$$. This is different from trail because an edge could be visited later. One could call such a walk as a no-turn-back walk. This type of walk could be of use in studying locally constrained graph homomorphisms.

Are no-turn-back walks studied in the literature?

Note: For simple graphs, the condition implies that no vertex is visited in the 'next step' itself.

• Yes. The common term is nonbacktracking walk Jan 18 at 2:51
• Okay, thank you. Jan 18 at 3:52