Activity preserving bijections between spanning trees and orientations in graphs

The main results of the paper are two dual algorithms bijectively mapping the set of spanning trees with internal activity 1 and external activity 0 of an ordered graph onto the set of acyclic orientations with adjacent unique source and sink. More generally, these algorithms extend to an activity-preserving correspondence between spanning trees and orientations. For certain linear orderings of the edges, they also provide a bijection between spanning trees with external activity 0 and acyclic orientations with a given unique sink. This construction uses notably an active decomposition for orientations of a graph which extends the notion of components for acyclic orientations with unique given sink.