We continue the study, initiated in [3], of the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties and given minimum cardinality. Let E be the following set of properties of digraphs: E ={strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, m...

A homomorphism from an oriented graph G to an oriented graph H is a mapping φ from the set of vertices of G to the set of vertices of H such that −−−−−−→ φ(u)φ(v) is an arc in H whenever −→ uv is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (th...

Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NPhard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every val...

We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree T of D such that at least k vertices in T have the same out-degree as in D. We show that this problem is W[1]-hard even on the class of directed acyclic graphs. In the dual version, called Reduced D...

Let k and r be two integers with k ≥ 2 and k ≥ r ≥ 1. In this paper we show that (1) if a strongly connected digraph D contains no directed cycle of length 1 modulo k, then D is k-colorable; and (2) if a digraph D contains no directed cycle of length r modulo k, then D can be vertex-colored with k colors so that each color class induces an acyclic subdigraph in D. The first result gives an affi...

We consider the following Turán-type problem: given a fixed tournament H, what is the least integer t = t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t = t(Tn, H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several results o...

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number ...

Suppose one needs to change the direction of at least n edges of an n-vertex tournament T , in order to make it H-free. A standard application of the regularity method shows that in this case T contains at least f∗ H( )n h copies of H, where f∗ H is some tower-type function. It has long been observed that many graph/digraph problems become easier when assuming that the host graph is a tournamen...

We consider the problem to find a set X of vertices (or arcs) with |X| ≤ k in a given digraph G such that D = G − X is an acyclic digraph. In its generality, this is Directed Feedback Vertex Set or Directed Feedback Arc Set respectively. The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made. In this p...

The notion of a competition graph was introduced by J. E. Cohen in 1968. The competition graph C(D) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the...