Good Orientations Of Unions Of Edge-Disjoint Spanning Trees

In this paper, we exhibit connections between the following subjects: Tree packing in graphs and digraphs (both behave completely different), the rigidity matroid of a graph, Henneberg moves on trees, the conjectures of Thomassen and Matthews and Sumner, and (s,t)-orderings of digraphs. We do this by studying graphs which admit acyclic orientations that contain an out-branching and in-branching which are arc-disjoint (such an orientation is calledgood). A2T-graphis a graph whose edge set can be decomposed into two edge-disjoint spanning trees. It is a well-known result due to Tutte and Nash-Williams, respectively, that every 4-edge-connected graph contains a spanning 2T-graph. Vertex-minimal 2T-graphs with at least two vertices which are known asgeneric circuitsplay an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that ifGis 2T-graph whose vertex set has a partitionV1,V2, horizontal ellipsis ,Vkso that eachViinduces a generic circuitGiofGand the set of edges between differentGi's form a matching inG, thenGhas a good orientation. We also obtain a characterization for the case when the set of edges between differentGi's form adouble tree, that is, if we contract eachGito one vertex, and delete parallel edges we obtain a tree. All our proofs are constructive and imply polynomial algorithms for finding the desired good orderings and the pairs of arc-disjoint branchings which certify that the orderings are good. We identify a structure which can be used to certify that a given 2T-graph does not have a good orientation.