I am reading A Textbook of Graph Theory by Balakrishnan and Ranganathan. I believe there is a problem with the following exercises.
Exercise 3.7 (a) Show that a graph $G$ with at least three vertices is 2-connected if and only if any vertex and any edge of $G$ lie on a common cycle of $G$. (b) Show that a graph $G$ with at least three vertices is 2-connected if and only if any two edges of $G$ lie on a common cycle.
For a graph with at least three vertices, the definition of 2-connected can be: 'any two vertices lie on a common cycle'.
I don't think the converse direction of either of these statements is true. My counterexample is the graph of just three isolated vertices and no edges. Can somebody confirm this?
I think it would suffice to include the assumption that $G$ is connected for the converse statements to hold.