On finding orientations with fewest number of vartices with small   out-degree
  Given an undirected graph, each of the two end-vertices of an edge can own the edge. Call a vertex poor, if it owns at most one edge. We give a polynomial time algorithm for the problem of finding an assignment of owners to the edges which minimizes the number of poor vertices. In the terminology of graph orientation, this means finding an orientation for the edges of a graph minimizing the number of edges with out-degree at most 1, and answers a question of Asahiro Jansson, Miyano, Ono (2014). 
Kaveh Khoshkhah
cs.DM