Strongly connected DAG from any connected undirected graph? I have the following question. It seems likely to be true - can anyone provide a standard reference?
Given:
A connected, undirected graph.
Question 1:
Can we assume a single direction for each edge such that the resulting directed graph is acyclic and strongly connected (path exists in one direction)?
Question 2:
Suppose that if $i< j$ and there is an edge between $i$ and $j$ in the undirected graph, the edge in the directed graph is pointing from $i$ to $j$. Is there always a node numbering such that the resulting graph is strongly connected and acyclic? (e.g. would numbering down the branches of a spanning tree in the undirected graph work?)
 A: The answer to both questions is negative, if one wants to have an acyclic digraph that is strongly connected, in the sense that any two nodes have a directed path between them in one direction or the other. 
A counterexample is provided by any tree having at least three leaves. It must be that two of the leaves are sources, or two are sinks, and in either case, there is no directed path between those two leaves.
If you drop the connectivity requirement, as you have suggested, then it is easy: place any linear order on the vertices of the original graph, and simply orient the edges to conform with that order. This is called a grading. Since you asked for a reference, graded digraphs are used in Every model of set theory embeds into its own constructible universe, where I develop a little of the theory in section 2, but I assume this is a standard topic in graph theory and there must be other references. 
A: You can construct such a directed graph relatively easily (if I've understood your question correctly); choose one vertex, and "hang" the rest of the graph from that vertex (i.e. create a "distance function" that specifies how far a vertex is from your original vertex). Then give each edge a "downwards" direction (i.e. the direction of increasing distance); then every vertex is reachable from that original vertex.
If I've misunderstood your question, then test your question on the two connected 3-vertex graphs; that should give an answer.
