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?)