§ questions on the structure of graphs

I've been trying to get more of a feeling for graphs lately, so I'm collecting sources of "structural" questions of graphs and answers for these questions.

Q. Is it possible in an unweighted graph, there is exactly one unique shortest
path tree for a node u but more than one such shortest path tree for some
other node v ?

Q. Can I orient the edges of a bridge-less undirected bipartite graph with even
no. of nodes such that all the nodes are in a cycle ?