Discrete Mathematics Graph G containing cycle C

Question

Graph G contains cycle C. It is also known that every vertex, which isn't in cycle C, belongs in a chain, whose first or last vertex of a path is in cycle C.

Is that graph connected? How to explain that?

Best regards

Answer 1

Pick any two vertices $u_1$ and $u_2$.

Let $u_1$ be in the chain $C_1$ and $u_2$ be in $C_2$, and let $C_1$ have a vertex $v_1$ in the cycle, whilst $C_2$ has vertex $v_2$ in the cycle.

Then $u_1,\dots \; (C_1) \; \dots ,v_1,\dots \; (C) \; \dots v_2,\dots \; (C_2) \dots ,u_2$ is a $uv$-walk.

Hence the graph is conncected