For a digraph D, let λ(D) be the arc-strong-connectivity of D. For an integer k > 0, a simple digraph Dwith |V (D)| ≥ k + 1 is k-maximal if every subdigraph H of D satisfies λ(H) ≤ k but for adding new arc to D results in a subdigraph H ′ with λ(H ) ≥ k + 1. We prove that if D is a simple k-maximal digraph on n > k + 1 ≥ 2 vertices, then |A(D)| ≥ n 2  + (n − 1)k +  n k + 2   1 + 2k −  k +...

Given a digraph D, let δ(D) := min{δ(D), δ−(D)} be the minimum degree of D. We show that every sufficiently large digraph D with δ(D) ≥ n/2 + l − 1 is l-linked. The bound on the minimum degree is best possible and confirms a conjecture of Manoussakis [16]. We also determine the smallest minimum degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every sequence s...

The main concern of this article is to present and motivate the Rubis method for tackling the choice problem in the context of multiple criteria decision aiding. Its genuine purpose is to help a decision maker to determine a single best decision alternative. Methodologically we focus on pairwise comparisons of these alternatives which lead to the concept of bipolar-valued outranking digraph. Th...

A semicomplete multipartite or semicomplete c-partite digraph D is a biorientation of a c-partite graph. A semicomplete multipartite digraph D is called strongly quasiHamiltonian-connected, if for any two distinct vertices x and y of D, there is a path P from x to y such that P contains at least one vertex from each partite set of D. In this paper we show that every 4-strong semicomplete multip...

An n-lift of a digraph K, is a digraph with vertex set V (K)× [n] and for each directed edge (i, j) ∈ E(K) there is a perfect matching between fibers {i} × [n] and {j} × [n], with edges directed from fiber i to fiber j. If these matchings are chosen independently and uniformly at random then we say that we have a random n-lift. We show that if h is sufficiently large then a random n-lift of the...

Suppose that D = (V,E) is a strongly connected digraph. Let u, v ∈ V (D). The maximum distance md(u, v) is defined as md(u, v)=max{ −→ d (u, v), −→ d (v, u)} where −→ d (u, v) denote the length of a shortest directed u− v path in D. This is a metric. The boundary, contour, eccentric and peripheral sets of a strong digraph D are defined with respect to this metric. The main aim of this paper is ...

The base of a signed digraph S is the minimum number k such that for any vertices u, v of S, there is a pair of walks of length k from u to v with different signs. Let K be a signed complete graph of order n, which is a signed digraph obtained by assigning +1 or −1 to each arc of the n-th order complete graph Kn considered as a digraph. In this paper we show that for n ≥ 3 the base of a primiti...

For the complete digraph DKn with n¿3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph DK5 in four of the six possible cases. For n = 9 + 6k; k = 0; 1; : : : ; a third part D ...

A paired comparison digraph D is a weighted digraph in which the sum of the weights of arcs, if any, joining two vertices exactly one. A one-to-one mapping from V (D) onto {1, 2, . . . , |V (D)|} is called a ranking ofD, and a ranking α ofD is optimal if the backward length of α is minimum. We say that D is r-partite if V (D) can be partitioned into V1 ∪ · · · ∪ Vr so that every arc of D joinin...

A signed digraph S is the digraph D by assigning signs 1 or −1 to each arc of D. The base of S is the minimum number k such that there is a pair walks which have the same initial and terminal point with length k, but different signs. In this paper we show that for n ≥ 5 the upper bound of the base of a primitive non-powerful signed tournament Sn, which is the signed digraph by assigning 1 or −1...