# Do these graphs belong to a known class of graphs?

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.

• Sorry, I put the images of my examples directly inside the question. but it seems that they cannot be displayed correctly, at least on my browser...! However, replacing images by the links to the images also did not help. – Toughee May 5 '17 at 5:16
• Not meaning to offend, but what is the purpose of this question? Do you want to know properties, or did you just come up with a random graph and wanted to see if someone had done it before. Loads of very obvious things can be said about such graphs, but this feels too arbitrary to have a name. – adfriedman May 5 '17 at 6:09
• @adfriedman Thanks for your attention. As you said I've just come up with them and I need to call them in general. – Toughee May 10 '17 at 23:34

I read about such graphs as lollipops in the context of Kotzig's conjecture on $P(k)$-graphs (not to be confused with the more famous conjecture of Kotzig on graceful labelings). I think it was mentioned in a paper of Bondy. But this is probably not very standard but more problem dependend.