graph theory conventions, difference between a PATH and a GRAPH?

Consider this example, I did my pseudocode in python

findCycle(G):     for each edge e in E(G):         isThereCycle(G-e):             G = G - e     return G  

assume isthereCycle returns true or false if graph $ G$ has a cycle.

Input is a graph $ G$ that contains a cycle and returns a path of that cycle.

We go through each edge then remove it and see if a cycle still consist without that edge, until we are left with a graph with a single path that is a cycle. (Assume removing the edge (G – e) doesn’t mutate the graph unless we do G = G – e).

I want to return a path. Is what I’m returning a path?