We call a digraph h-semicomplete if each vertex of the digraph has at most h non-neighbors, where a non-neighbor of a vertex v is a vertex u 6= v such that there is no edge between u and v in either direction. This notion generalizes that of semicomplete digraphs which are 0-semicomplete and tournaments which are semicomplete and have no anti-parallel pairs of edges. Our results in this paper a...

The Optical Transpose Interconnection System (OTIS) was proposed by Marsden et al. [Opt. Lett 18 (1993) 1083–1085] to implement very dense one-to-one interconnection between processors in a free space of optical interconnections. The system which allows one-to-one optical communications from p groups of q transmitters to q groups of p receivers, using electronic intragroup communications for ea...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1, 2, . . . , m} where m ≥ 1. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if there is no monochromatic path...

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynom...

Let D be a digraph and X ~ V(D). By pushing X we mean reversing the orientation of each arc of D with exactly one end in X. Klostermeyer proved that it is NP-complete to decide if a given digraph can be made acyclic using the push operation. By contrast, Huang, MacGillivray, and Wood showed that the problem of deciding if a given multipartite tournament can be made acyclic using the push operat...

It is known that the spectral radius of a digraph with k edges is at most √ k, and that this inequality is strict except when k is a perfect square. For k = m2 + , fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex, a corresponding loop, and then connecting it to the first /2 vertices by pairs of directed edge...

We construct an efficient randomized algorithm that, given a digraph, tests if it is a strongly connected graph. And the query complexity of our algorithm is polylogarithmic in the size of the digraph.