Consider the cycle $C_3$ connected to the path $P_4$, on just one of its vertices. Does this graph have a special name?
or for example:
More generally, for $\;m\geq2\;,\;n\geq3$ , is there a (minimal) class of graphs that contains the $n$-cycles $C_n$ with one of their vertices connected to a path $P_m$?
Any useful notes would be appreciated.