In graph theory, walks can be described with these terms according to the glossary of graph theoretic terms, wikipedia, and this page.
- walk: an alternating sequence of vertices and edges, starting and ending at a vertex, in which each edge is adjacent in the sequence to its two endpoints
- trek: a walk that does not backtrack, i.e. no two successive edges are the same
- trail: a trek where all edges are distinct
- path: a trail where all vertices are distinct (except maybe the first and last)
I am trying to see if there is there are specific names for the closed versions, ie when the first and last vertices are the same:
- (1) closed walk
- (2) closed trek
- (3) closed trail
- (4) closed path
- (5) plus the term corresponding to the non-specification of the "first" and "last" vertex (because it's a cycle, it does not "matter" where we start ie the thing called a circuit on this page)
So far I have read these terms:
- cycle
- elementary cycle
- circuit
- tour
- ...
but I do not know which one correspond to (1), (2), (3), (4) and (5).
Question: So in short, what are the most commonly accepted mathematical terms (if they exist) to designate (1), (2), (3), (4) and (5)?
Closed walk: cycle.
Closed path: simple cycle, a.k.a. elementary cycle.
Closed trail: circuit.
Tour is just a synonym for trek.
I have never heard of a shortcut for closed tour. Also, I am not convinced that this definition of circuit (where the initial point is not specified) is universally accepted. I have never encountered a text where the authors would take for granted that such or such words would imply that the initial point doesn't matter. If needed, they would specify it.