Make a graph singly connected by edge orientations
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A directed graph D is singly connected if for every ordered pair of vertices (s,t), there is at most one path from s to t in D. Graph orientation problems ask, given an undirected graph G, to find an orientation of the edges such that the resultant directed graph D has a certain property. In this work, we study the graph orientation problem where the desired property is that D is singly connected. Our main result concerns graphs of a fixed girth g and coloring number c. For every g,c≥ 3, the problem restricted to instances of girth g and coloring number c, is either NP-complete or in P. As further algorithmic results, we show that the problem is NP-hard on planar graphs and polynomial time solvable distance-hereditary graphs.
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