Packing directed circuits exactly

Graphs and digraphs in this paper may have loops and multiple edges. Paths and circuits have no “repeated” vertices, and in digraphs they are directed. A transversal in a digraph D is a set of vertices T which intersects every circuit, i.e. DnT is acyclic. A packing of circuits (or packing for short) is a collection of pairwise (vertex-)disjoint circuits. The cardinality of a minimum transversal is denoted by (D) and the cardinality of a maximum packing is denoted by (D). Clearly (D) (D), and our objective is to study when equality holds. We will show in Section 4 that this is the case for every strongly planar digraph. (A digraph is strongly planar if it has a planar drawing such that for every vertex v, the edges with head v form an interval in the cyclic ordering of edges incident with v.) However, in general there is probably no nice characterization of digraphs for which equality holds, and so instead we characterize digraphs such that equality holds for every subdigraph. Thus we say that a digraph D packs if (D0) = (D0) for every subdigraph D0 of D. We will give two characterizations: one in terms of excluded minors, and the other will give a structural description of digraphs that pack. We say that an edge e of a digraph D with head v and tail u is special if either e is the only edge of D with head v, or it is the only edge of D with tail u, or both. We say that a digraphD is a minor of a digraphD0 ifD can be obtained from a subdigraph of D0 by repeatedly contracting special edges. It is easy to see that if a digraph packs, then so do all its minors. Thus digraphs that pack can be characterized by a list of minor-minimal digraphs that do not pack. By an odd double circuit we mean the digraph obtained from an undirected circuit of odd length at least three by replacing each edge by a pair of directed edges, one in each direction. The digraph F7 is defined in Figure 1. The following is our excluded minor characterization.