Let D = (V,A) be a digraph of order n, S a subset of V of size k and 2 ≤ k ≤ n. Strong subgraphs D1, . . . , Dp containing S are said to be internally disjoint if V (Di)∩V (Dj) = S and A(Di)∩A(Dj) = ∅ for all 1 ≤ i < j ≤ p. Let κS(D) be the maximum number of internally disjoint strong digraphs containing S in D. The strong subgraph kconnectivity is defined as κk(D) = min{κS(D) | S ⊆ V, |S| = k}...

We have recently proposed a robustified ratio consensus algorithm which achieves asymptotic convergence to the global average in a distributed fashion in static strongly connected digraphs, despite the possible presence of bounded but otherwise arbitrary delays. In this work, we propose a protocol which reaches asymptotic convergence to the global average in a distributed fashion under possible...

A strongly connected digraph D of order n is primitive (aperiodic) provided the greatest common divisor of its directed cycle lengths equals 1. For such a digraph there is a minimum integer t, called the exponent of D, such that given any ordered pair of vertices x and y there is a directed walk from x to y of length t. The exponent of D is the largest of n ‘generalized exponents’ that may be a...

A star is an arborescence in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. The directed star arboricity of a digraph D, denoted by dst(D), is the minimum number of galaxies needed to cover A(D). In this paper, we show that dst(D) ≤ Δ(D) + 1 and that if D is acyclic then dst(D) ≤ Δ(D). These results are proved by considering the existence of spann...

We solve the Firing Squad Synchronization Problem (FSSP), for P systems based on digraphs with simplex channels, where communication is restricted by the direction of structural arcs. Previous work on FSSP for P systems focused exclusively on P systems with duplex channels, where communication between parents and children is bidirectional. Our P solution, the first for simplex channels, require...

The descendant set desc(α) of a vertex α in a digraph D is the set of vertices which can be reached by a directed path from α. A subdigraph of D is finitely generated if it is the union of finitely many descendant sets and D is descendant-homogeneous if it is vertex transitive and any isomorphism between finitely generated subdigraphs extends to an automorphism. We consider connected descendant...