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Whenever there is a back edge, a cycle is detected. Then, do a pair of parallel edges form a cycle in a graph? If no, why it is not a back edge?

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Graph theory is notorious for having nonequivalent definitions of the same term being used by different authors in different places, so it is important to check the definition of the terms in your case. If you use the definition I'm familiar with \begin{align*} &\text{A }\textit{cycle }\mathcal{C}\text{ is an alternating sequence of vertices and edges, beginning with a vertex }v_0\\ &\text{ such that no vertices or edges repeat except that }v_0\text{is both the first and last element}\\ &\text{of the sequence} \end{align*} In that case if $G=(V,E,f)$, $v_1,v_2\in V$, $e_1,e_2\in E$ and $f(e_1) = f(e_2) = \{v_1,v_2\}$ then $v_1e_1v_2e_2v_1$ is a cycle in $G$.

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