Good orientations of unions of edge‐disjoint spanning trees - Bang‐Jensen, Jørgen; Bessy, Stéphane; Huang, Jing; Kriesell, Matthias (Journal of graph theory, 2021)
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 called good). A 2T‐graph is 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 as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if G is 2T‐graph whose vertex set has a partition V1,V2,…,Vk so that each Vi induces a generic circuit Gi of G and the set of edges between different Gi's form a matching in G, then G has a good orientation. We also obtain a characterization for the case when the set of edges between different Gi's form a double tree, that is, if we contract each Gi to 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.