In this communication, we introduce an original algorithm for computing both dominant and absorbent kernels in a bipolar ordinal valued digraph. The approach relies on theoretical results recently obtained by Bisdorff, Pirlot and Roubens [8] which give a constructive – fixpoint equation based – proof of the bijection between bipolar ordinal valued kernels of such a valued digraph and the crisp ...

Consider a directed graph (digraph) in which vertices are assigned color sets, and two connected if only they share at least one the tail vertex has strictly smaller set than head. We seek to determine smallest possible size of union sets that allows for such digraph representation. To address this problem, we introduce new notion intersection representation digraph, show it is well-defined all...

For a given $2$-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $B_G(V_1,V_2)$ of $G$ induced by those arcs/edges that have one end in each $V_i$. We determine, for all pairs of non-negative integers $k_1,k_2$, the complexity of deciding whether $G$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_i$ has at least $k_i$ ...

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...