In chemical graph theory or chemical-related topology a distinction is made between a cycle, a ring, and a strong ring. It is relatively easy to define a cycle:

**A cycle is a continuous path (along edges) in which the start and end vertex are the same.**

So this means a cycle (also sometimes called a circuit) can be of any size, regardless if there are shorter (or also longer) paths to get back to the ‘home vertex’. We can also say that a cycle is a closed path.

What might be a little more difficult to understand is the formal definition of a ring:

**A ring is a cycle that is not the sum of any two smaller cycles.**

It is important to emphasize *both* attributes: two *and* smaller. The first conclusion is: Okey, in a graph or net there might be rings that are the sum of three, four, five (smaller) cycles, but a cycle is not a ring if it is the sum of exactly two (smaller) cycles.

In the following figure the graph of a cube is shown and some cycles (highlighted in red) within this graph:

Which of these cycles are rings? I think it is clear, that the two cycles in the top row are rings. Furthermore, the cycle in the middle in the bottom row is a ring, while the left and right cycle are not rings. Why? Well, the 6-cycle on the left is the sum of two smaller 4-cycles. The 6-cycle in the middle is the sum of three(!) smaller cycles but not the sum of two smaller cycles. And the 8-cycle on the right can be considered as the sum of a 4-cycle and a 6-cycle (the latter can be formed as the sum of two smaller 4-cycles, the top-most and the central one):

Well, but shouldn’t it be possible to build the 6-cycle in the middle as the sum of a 4-cycle (top-most) and a 6-cycle (which itself be the sum of the central and right 4-cycles)? Yes, that is correct, but the definition states that the cycles of a sum of cycles have to be smaller! This is not the case here!

Interestingly, there is an alternative definition of a ring which is, at least for me, much more intuitive and comprehensible:

**A ring is a cycle that contains along its path no potential short-cut (of the length of exactly one edge) to the ‘home vertex’.**

It is easy to see that the cycle on the left has one short-cut and the cycle on the right even two (dashed edges):

So, what are now *strong* rings? They can be defined as follows:

**Strong rings are rings that are not the sum of any number of smaller cycles.**

From that definition It follows that only the two cycles in the top row are strong rings.

I would like to thank Michael Fischer (@ZeoliteMiFi) for helpful discussions.