In my computer science background, I have thought that cycles can contain other cycles in a graph, as they have been defined for me in cs classes merely as a path which starts an ends at the same vertex. However, I am presented with this algebraic definition of a cycle:
A cycle (or circuit) is a sequence of edges $\left\{v_1, v_2\right\},\left\{v_2, v_3\right\}, \ldots,\left\{v_{n-2}, v_{n-1}\right\},\left\{v_{n-1}, v_n\right\},\left\{v_n, v_1\right\}$, where $v_1, \ldots, v_n$ are distinct.
Due to the variables representing these vertices being distinct, there can be no cycle within a cycle. I am wondering whether this definition precluding a cycle from being within a cycle is the standard one in discrete mathematics and graph theory, as it clashes with my cs background.
Best Answer
Usually you should check definitions in the article or textbook. Different authors give different meaning to the same terms. According to Bondy, Murty and their Graph theory with applications, cycle is a closed trail with no repeated vertex other than origin and terminus, i. e. cycle is a closed path.
On the other hand some authors say path instead of trail and simple path instead of path. Then the same happens with cycles: they use cycle for any closed trail and simple cycle for cycle.
Also it is possible to define cycle as a subgraph rather than as a sequence. But still the same difference occurs: some authors say that $2$-regular connected subgraph is a cycle, and some others say that it is a simple cycle.