A Playfair cipher is a digraph substitution, where each letter pair of the plaintext is replaced by a letter pair in the ciphertext. The chart of substitutions is summarized by the placement of letters in a table, called the Playfair square, constructed in some customary manner from a keyword. We denote a substitution of plaintext digraph cd with ciphertext digraph ab by ab− cd. There are two r...

The Shapley value for directed graph (digraph) games, TU games with limited cooperation introduced by an arbitrary digraph prescribing the dominance relation among the players, is introduced. It is defined as the average of marginal contribution vectors corresponding to all permutations that do not violate the subordination of players. We assume that in order to cooperate players may join only ...

In 1981, Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k − 1 contains k disjoint cycles. This conjecture is trivial for k = 1, and was established for k = 2 by Thomassen in 1983. We verify it for the next case, by proving that every digraph with minimum out-degree at least five contains three disjoint cycles. To show this, we improve Thomassen’s result b...

Manoussakis, Y. and D. Amar, Hamiltonian paths and cycles, number of arcs and independence number in digraphs, Discrete Mathematics 105 (1992) 157-172. We let D denote a digraph with n vertices, independence number at least (Y and half-degrees at least k. We give (i) a function f(n, a) (respectively f (n, k, a)) such that any digraph with at least f(n, a) (respectively f(n, k, a)) arcs is Hamil...

LetD = (V,A) be a finite and simple digraph. A Roman dominating function (RDF) on a digraph D is a labeling f : V (D) → {0, 1, 2} such that every vertex with label 0 has a in-neighbor with label 2. The weight of an RDF f is the value ω(f) = ∑ v∈V f(v). The Roman domination number of a digraph D, denoted by γR(D), equals the minimum weight of an RDF on D. In this paper we present some sharp boun...

Let F be a cellular automaton on the space of all sequences from a finite alphabet. From the local description of F a finite labeled digraph is constructed with a distinguished subdi graph. The surjectivity and openness of F are proved to correspond to properties of these digraphs. Similarly, another finite labeled digraph is constructed with a distin guished subset of its vert ex set . Th e in...

Abstract. A real n × n matrix is a Q-matrix if for every k = 1, 2, . . . , n the sum of all k × k principal minors is positive. A digraph D is said to have Q-completion if every partial Q-matrix specifying D can be completed to a Q-matrix. For the Q-completion problem, sufficient conditions for a digraph to have Q-completion are given, necessary conditions for a digraph to have Q-completion are...

The upward planarity testing problem consists of testing if a digraph admits a drawing Γ such that all edges in Γ are monotonically increasing in the vertical direction and no edges in Γ cross. In this paper we reduce the problem of testing a digraph for upward planarity to the problem of testing if its blocks admit upward planar drawings with certain properties. We also show how to test if a b...

A vertex set X of a digraph D = (V, A) is a kernel if X is independent (i.e., all pairs of distinct vertices of X are non-adjacent) and for every v ∈ V − X there exists x ∈ X such that vx ∈ A. A vertex set X of a digraph D = (V, A) is a quasi-kernel if X is independent and for every v ∈ V − X there exist w ∈ V − X, x ∈ X such that either vx ∈ A or vw, wx ∈ A. In 1994, Chvátal and Lovász proved ...

In this article, we present a detailed study of the reach distance-layer structure De Bruijn and Kautz digraphs, apply our analysis to performance evaluation deflection routing in networks. Concerning structure, provide explicit polynomial expressions, terms degree digraph, for cardinalities some relevant sets structure. Regarding application defection routing, as consequence description formul...