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, connected, minimum outdegree at least 1, minimum in-deg...

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 virus is a local configuration that, if present in a graph or a digraph, forbids these graphs or digraphs to have a specific property. The aim of this article is to sketch the evolution of the virus theory from its birth in 1991. Moreover some new results and open questions are given. The properties with its known viruses, that will be discussed in this work, are the following: hamiltonian, t...

We prove that connected vertex-transitive digraphs of order p5 (where p is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup G of prime power order such that G′ is generated by two elements or elementary abelian is Hamiltonian.

Galeana-Sanchez, H., On the existence of kernels and h-kernels in directed graphs, Discrete Mathematics 110 (1992) 2.51-255. A directed graph D with vertex set V is called cyclically h-partite (h > 2) provided one can partition V = V, + V, +. . + V,_, so that if (u, u) is an arc of D then u E V,, and u E v+, (notation mod h). In this communication we obtain a characterization of cyclically h-pa...

The energy of a digraph D with eigenvalues z1, z2, . . . , zn is defined as E(D) = n ∑ j=1 |Rzj |, where Rzj is the real part of the complex number zj . In this paper, we characterize some positive reals that cannot be the energy of a digraph. We also obtain a sharp lower bound for the energy of strongly connected digraphs.

A bipartite random mapping TK,L of a finite set V = V1 ∪ V2, |V1| = K and |V2| = L , into itself assigns independently to each i ∈ V1 its unique image j ∈ V2 with probability 1/L and to each i ∈ V2 its unique image j ∈ V1 with probability 1/K . We study the connected component structure of a random digraph G(TK,L) , representing TK,L, as K → ∞ and L → ∞. We show that, no matter how K and L tend...

We derive an asymptotic formula for the number of strongly connected digraphs with n vertices and m arcs (directed edges), valid for m − n → ∞ as n → ∞ provided m = O(n log n). This fills the gap between Wright’s results which apply to m = n+O(1), and the long-known threshold for m, above which a random digraph with n vertices and m arcs is likely to be strongly connected.

A digraph without loops, multiple arcs and directed cycles of length two is called an in-tournament if the set of in-neighbors of every vertex induces a tournament. A local tournament is an in-tournament such that the set of out-neighbors of every vertex induces a tournament as well. Let p ≥ 2 be an integer and let T be a strongly connected tournament such that every vertex has at least p posit...