Let $T$ be a given (finite) tree.


**Question 1:** Is it always possible to add edges to $T$ to obtain a $2$-connected outerplanar supergraph $G$?


**Question 2:** If the answer to Question #1 is negative, can the trees for which it is possible be characterized?


**Question 3( Defect form of Question 1):** Let $T$ be a *rooted* tree with root vertex $v\_{0} \in V(T)$. Is it always possible to add edges to $T$ to obtain a $2$-connected planar graph $G$ with a plane embedding in which $v\_{0}$ is the only internal vertex?



Completing a tree to a 2-connected outerplanar graph
Yes. Pick an arbitrary vertex to be the root. Consider the sequence of vertices $v\_1, v\_2, \ldots$ produced by a pre-order traversal of the rooted tree, adding edges $v\_i - v\_{i+1}$ where they don't already exist. Finally, add an edge back from the last vertex to the root, if it doesn't already exist. This cycle defines the outer face and gives 2-connectivity.


