A digraph D is said to be an m-coloured digraph, if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) of vertices of D is said to be a kernel by monochromatic paths of the m-coloured digraph D, if it satisfies the two following properties: (1) N is independent by monochromatic paths; i.e., f...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) −N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k − 1)-kernel. For a strong digraph D, a set S ⊂ V (D) is a separator if D \ S is not strong, D is σ-...

We describe DiGraph, a customisable graph editor that has been built using the Tcl/Tk programming system. We show how DiGraph can be customised to act as a programming environment for the Hathi-2 trans-puter machine. We also discuss how DiGraph could be used as a base for building programming environments to other parallel/distributed systems that use point-to-point message passing.

A digraph is said to be super-connected if every minimum vertex cut is the out-neighbor set or in-neighbor set of a vertex. A digraph is said to be reducible, if there are two vertices with the same out-neighbor set or the same in-neighbor set. In this paper, we prove that a strongly connected arc-transitive oriented graph is either reducible or super-connected. Furthermore, if this digraph is ...

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V (D) − N there exists an arc from w to N . The digraph D is said to be a kernel-perfect digraph when every induced subdigraph of D has a kernel. Minimal non kernel-perfect digraphs are called critical kernel imperfect digraphs. In this paper some new structural results concerning finite critical kernel imp...

We investigate sufficient conditions, and in case that D be an asymmetrical digraph a necessary and sufficient condition for a digraph to have the following property: “In any induced subdigraph H of D, every maximal independent set meets every non-augmentable path”. Also we obtain a necessary and sufficient condition for any orientation of a graph G results a digraph with the above property. Th...

A digraph with maximum out-degree d and radius k has at most 1+d+ · · ·+d vertices, as the Moore bound states. Regular digraphs attaining such a bound and whose diameter is at most k + 1 are called radially Moore digraphs. Knor [4] proved that these extremal digraphs do exist for any value of d ≥ 1 and k ≥ 1. In this paper, we introduce a digraph operator, based on the line digraph, which allow...

A weighted composite digraph is obtained from some weighted digraph by replacing each vertex with a weighted digraph. In this paper, we give a beautiful combinatorial proof of the formula for forest volumes of composite digraphs obtained by Kelmans-Pak-Postnikov [7]. Moreover, a generalization of this formula is present.

Given a finite abelian group A, a subset ⊆ A and an endomorphism of A, the endo-Cayley digraph GA( , ) is defined by taking A as the vertex set and making every vertex x adjacent to the vertices (x)+ a with a ∈ . When A is cyclic and the set is of the form = {e, e + h, . . . , e + (d − 1)h}, the digraph G is called a consecutive digraph. In this paper we study the hamiltonicity of endo-Cayley d...