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If $G$ and $H$ are graphs with vertex set $V(G), V(H)$ and edge set $E(G), E(H)$ respectively. Define the sum $G+H$ to be the graph with vertex set $V(G) \cup V(H)$ and edge set $E(G) \cup E(H)$

How many unlabelled isomorphism classes of graphs are there on $n$ vertices which are a sum of cycle graphs on more than $3$ vertices?

So for example there are $2$ such graphs on $6$ vertices namely $C_6$ and $C_3 + C_3$. There are $4$ on $9$ vertices: $C_9$, $C_3+C_6$, $C_4 + C_5$ and $C_3 + C_3 + C_3$.

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To put it more simply, you are asking for the number of unlabelled $2$-regular graphs on $n$ vertices; equivalently, the number of partitions of $n$ into parts $\ge3.$ This is OEIS sequence A008483.

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