Covering a Strong Digraph by -1 Disjoint Paths: A Proof of Las Vergnas' Conjecture

The Gallai-Milgram theorem states that every directed graph D is spanned by α(D) disjoint directed paths, where α(D) is the size of a largest stable set of D. When α(D) > 1 and D is strongly connected, it has been conjectured by Las Vergnas (cited in [1] and [2]) that D is spanned by an arborescence with α(D)− 1 leaves. The case α = 2 follows from a result of Chen and Manalastas [5] (see also Bondy [3]). We give a proof of this conjecture in the general case. In this paper, loops, cycles of length two and multiple arcs are allowed. We denote by α(D) the stability number (or independence number) of D, that is, the cardinality of a largest stable set of D. A k-path partition P of a digraph D is a partition of the vertex set of D into k directed paths. A functional digraph is a digraph in which every vertex has indegree one. An arborescence is a connected digraph in which every vertex has indegree one except the root, which has indegree zero. The vertices of an arborescence (or a functional digraph) with outdegree zero are the leaves. An arborescence forest F is a disjoint union of arborescences. We denote by R(F ) the set of roots of the arborescences of F , and by L(F ) the set of its leaves. A strong component of D is a maximal strongly connected subgraph of D. A strong component C of D is maximal (resp. minimal) if no vertex of C has an out-neighbour (resp. in-neighbour) in D \ C. Theorem 1 (Las Vergnas [7], see also Berge [1]) Let D be a digraph, m1, . . . ,ml the minimal strong components of D and x1, . . . , xl vertices of m1, . . . ,ml, respectively. There exists a spanning arborescence forest F of D with R(F ) = {x1, . . . , xl} and |L(F )| ≤ α(D). Proof. First observe that there exists a spanning arborescence forest F of D with R(F ) = {x1, . . . , xl}. Now let us prove that if a spanning arborescence forest F of D with R(F ) = {x1, . . . , xl} has more than α(D) leaves, there exists a spanning arborescence forest F ′ of D with R(F ′) = {x1, . . . , xl}, |L(F ′)| = |L(F )| − 1 and L(F ′) ⊂ L(F ). Such a forest F ′ is a reduction of F . This statement is easily proved by induction on D: Since |L(F )| > α(D), there exist two leaves x, y of F such that xy ∈ E(D). Apply a reduction to D \ y and F \ y, and add y to this reduction in order to conclude. To prove Theorem 1, apply successive reductions to a spanning arborescence forest F of D with R(F ) = {x1, . . . , xl}. Corollary 1.1 (Gallai and Milgram [6]) Every digraph D admits an α(D)-path partition. We now prove that every strong digraph with stability number α > 1 is spanned by an arborescence with α − 1 leaves. This answers a question of Las Vergnas (cited in [1] and [2]) and extends a result of