A (k1 + k2)-bispindle is the union of k1 (x, y)-dipaths and k2 (y, x)-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every (1, 1)bispindle B, there exists an integer k such that every strongly connected digraph with chromatic number greater than k contains a subdivision of B. We investigate generalisations of this result by first showing co...

The problem of the existence of an alternating simple dicycle in a 2-arc-coloured digraph is considered. This is a generalization of the alternating cycle problem in 2-edgecoloured graphs (proved to be polynomial time solvable) and the even dicycle problem (the complexity is not known yet). We prove that the alternating dicycle problem is NPcomplete. Let f(n) (g(n), resp.) be the minimum intege...

In this paper, we prove the following sufficient condition for the existence of k-kernels in digraphs: Let D be a digraph whose asymmetrical part is strongly conneted and such that every directed triangle has at least two symmetrical arcs. If every directed cycle γ of D with `(γ) 6≡ 0(mod k), k ≥ 2 satisfies at least one of the following properties: (a) γ has two symmetrical arcs, (b) γ has fou...

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...

We consider strongly connected networks of input-to-state stable (ISS) systems. Provided a small gain condition holds it is shown how to construct an ISS Lyapunov function using ISS Lyapunov functions of the subsystems. The construction relies on two steps: The construction of a strictly increasing path in a region defined on the positive orthant in R by the gain matrix and the combination of t...

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...

Let A = (aij ) ∈ Rn×n,N = {1, . . . , n} and DA be the digraph (N, {(i, j); aij > −∞}). The matrix A is called irreducible if DA is strongly connected, and strongly irreducible if every maxalgebraic power of A is irreducible. A is called robust if for every x with at least one finite component, A(k) ⊗ x is an eigenvector of A for some natural number k. We study the eigenvalue–eigenvector proble...

In this paper, we completely determine the connectivity of every infinite circulant digraphs and prove that almost all infinite circulant digraphs are infinitely strongly connected and therefore have both oneand two-way infinite Hamiltonian paths.

For designing reliable and efficient communications networks, the problem of constructing a maximally connected d-regular digraph (directed graph) with a small diameter is investigated. A maximally connected d-regular digraph with a diameter at most two larger than the lower bound for any number of nodes n > 2d and any d > 3 is constructed. Since the diameter of this digraph is quasiminimal (at...