BANG-JENSEN , Jørgen

We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected,...

International audience

We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected,...

A digraph is eulerian if it is connected and every vertex has its in-degree equal to its outdegree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph containing a....

The concept of arc-disjoint flows in networks was recently introduced in \cite{bangTCSflow}. This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source $s$ to all other...

A digraph is eulerian if it is connected and every vertex has its in-degree equal to its outdegree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph containing a....

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $H$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial...

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) G, does it contain an induced subdivision of a prescribed digraph D? The complexity of this problem depends on D and on whether G must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial...

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 EE be the following set of properties of digraphs: E=E={strongly connected, connected, minimum out-degree at least 1, minimum...

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) H, does it contain an induced subdivision of a prescribed digraph D? The complexity of this problem depends on D and on whether H is an oriented graph or contains 2-cycles. We announce a number of examples of polynomial instances as well as...

We continue the study, initiated in INRIA research report RR-8867, 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 con- nected, connected, minimum...

The concept of arc-disjoint flows in networks was recently introduced in Bang-Jensen and Bessy (2014). This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source s to all other...

The support of a flow $x$ in a network is the subdigraph induced by the arcs $ij$ for which $x_{ij}>0$. We discuss a number of results on flows in networks where we put certain restrictions on structure of the support of the flow. Many of these problems are NP-hard because they generalize linkage problems for digraphs. For example deciding...

For a given 2-partition (V1, V2) of the vertices of a (di)graph G, we study properties of the spanning bipartite subdigraph BG(V1, V2) of G induced by those arcs/edges that have one end in each Vi, i ∈ {1, 2}. We determine, for all pairs of non-negative integers k1, k2, the complexity of deciding whether G has a 2-partition (V1, V2) such that...

Let D be an oriented graph. The inversion of a set X of vertices in D consists in reversing the direction of all arcs with both ends in X. The inversion number of D, denoted by inv(D), is the minimum number of inversions needed to make D acyclic. Denoting by τ (D), τ (D), and ν(D) the cycle transversal number, the cycle arc-transversal number...

Let k be a fixed integer. We determine the complexity of finding a p-partition (V1,. .. , Vp) of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by Vi, (1 ≤ i ≤ p) is at least k smaller than the maximum out-degree of D. We show that this problem is polynomial-time solvable when p ≥ 2k and N...

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We prove that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B+$ and an in-branching $B−$ which are arc-disjoint (we call such branchings a good pair). This is best possible in terms of the arc-connectivity as there are infinitely many strong digraphs with independence number 2 and...

We consider the following problem for oriented graphs and digraphs: Given a directed graph D, does it contain a subdivision of a prescribed digraph F? We give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems.

We study the complexity of deciding whether a given digraph D = (V, A) admits a partition (A1, A2) of its arc set such that each of the corresponding digraphs D1 = (V, A1) and D2 = (V, A2) satisfy some given prescribed property. We mainly focus on the following 15 properties: being bipartite, being connected, being strongly connected, being...

We consider the following problem for oriented graphs and digraphs: Given a directed graph D, does it contain a subdivision of a prescribed digraph F? We give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems.

A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every...

A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every...