What do we call a closed path?

A closed path is referred to as a cycle. In the context of graph theory, a cycle is defined as a path that begins and ends at the same vertex without traversing any edge more than once. This characteristic of returning to the starting point while ensuring that the path does not overlap allows for a clear classification of the path as a cycle.

In contrast, while terms like loop and trail may be associated with closed or particular types of paths, they do not universally encapsulate the idea of returning to the original vertex without edge repetition. A loop typically refers to a single edge connected at both ends to the same vertex, which is not necessarily a closed path in the sense of traversing multiple edges. A circuit generally implies a closed path in the context of electricity or specific structured networks, but it is not limited to the mathematical definition applicable in graph theory. A trail, on the other hand, can be a path in which vertices can be repeated but edges cannot, which again does not fit the specific definition of a cycle.

Thus, the most accurate term for a closed path in the mathematical sense is a cycle.