## Abstract

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 (Formula presented.) is 2T-graph whose vertex set has a partition (Formula presented.) so that each (Formula presented.) induces a generic circuit (Formula presented.) of (Formula presented.) and the set of edges between different (Formula presented.) 's form a matching in (Formula presented.), then (Formula presented.) has a good orientation. We also obtain a characterization for the case when the set of edges between different (Formula presented.) 's form a double tree, that is, if we contract each (Formula presented.) 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.

Original language | English |
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Journal | Journal of Graph Theory |

Volume | 96 |

Issue number | 4 |

Pages (from-to) | 594-618 |

ISSN | 0364-9024 |

DOIs | |

Publication status | Published - Mar 2021 |

## Keywords

- 2T-graph
- acyclic digraph
- acyclic orientation
- branchings
- generic circuit
- NP-complete problem
- polynomial algorithm
- rigidity matroid
- spanning trees
- vertex ordering