The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) −...

A digraph is called k-cyclic if it cannot be made acyclic by removing less than k arcs. It is proved that for every > 0 there are constants K and δ so that for every d ∈ (0, δn), every n2-cyclic digraph with n vertices contains a directed cycle whose length is between d and d+K. A more general result of the same form is obtained for blow-ups of directed cycles.

Directed acyclic graphs provide a convenient representation of reticulate evolution in systematic biology. In this paper we formalize and analyse a simple model in which evolved characteristics are passed on to all descendant species. We show that the resulting observed sets of characteristics for the species at the leaves uniquely determine the digraph that described the evolution of the speci...

Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices nk of the k-iterated line digraph L (G), for k ≥ 0, where L(G) = G. We obtain this result by using the minimal polynomial of a quotient digraph π(G) of G. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our ...

A subgraph H of an acyclic digraph D is convex if there is no directed path between vertices of H which contains an arc not in H. A digraph D is connected if the underlying undirected graph of D is connected. We construct an algorithm for enumeration of all connected convex subgraphs of a connected acyclic digraph D of order n. The time complexity of the algorithm is O(n · cc(D)), where cc(D) i...

An upward topological book embedding of a planar st-digraph G is an upward planar drawing of G such that its vertices are aligned along the vertical line, called the spine, and each edge is represented as a simple Jordan curve which is divided by the intersections with the spine (spine crossings) into segments such that any two consecutive segments are located at opposite sides of the spine. Wh...

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f(k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f(5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root...

Let D be a digraph and let λ ( ) denote the number of vertices in longest path . For pair vertex-disjoint induced subdigraphs A B , we say that is partition if V ∪ = The Path Partition Conjecture (PPC) states for every digraph, integer q with 1 ≤ − there exists such T vertex set { u … t } i ∈ [ ] H j : n composition Q arc p We acyclic (semicomplete, respectively) respectively). In this paper, i...

A set X of vertices of an acyclic graph is convex if any vertex on a directed path between elements of X is itself in X. We construct an algorithm for generating all input-output constrained convex (IOCC) sets in an acyclic digraph, which uses several novel ideas. We show that our algorithm is more efficient than algorithms described in the literature in both the worst case and computational ex...

In the usual Gn;p{model of a random acyclic digraph let n (1) be the size of the reeexive, transitive closure of node 1, a source node; then the distribution of n (1) is given by where q = 1 ? p. Our analysis points out some surprising relations between this distribution and known functions of the number theory. In particular we nd for the expectation of n (1): lim n!1 n ? E(n (1)) = L(q) where...