Maximally connected digraphs

Abstract

This paper introduces a new parameter / = / ( G ) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let K, A, 6, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that A = 6 if D s 21 and K = 6 if D I 21 -1. Analogous results involving upper bounds for K and A are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-A and super-rc are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized. DIGRAPHS AND LINE DIGRAPHSThis paper concentrates on the connectivity of digraphs. More precisely, we study the relation between the connectivity, the diameter, and a new parameter that, in the case of graphs, is closely related with the girth. The results obtained have some interesting corollaries. For example, it is shown that iterated line digraphs are maximally connected if the iteration order is large enough.Let us first recall the notation used throughout the paper. Let G = ( K A ) denote a digraph with (finite) set of vertices V = V(G) and set of arcs A = A(G), which are ordered pairs of (not necessarily different) vertices of V. So, loops are allowed but parallel arcs are not. If e = (x, y ) E A, we say that x is adjacent to y and that y is ad'ucenrfrom x . Let r-(x) and I'+ (x) denote respectively the sets of vertices adjacent to and from x , i.e., the sets of in-neighbors and out-neighbors of x . Their cardinalities are the in-degree of x , K ( x ) = lI'-(x)l, and the out-degree of x , 6'(x) = Ir'(x)[. The minimum degree of G, 6 = 6(G), is the minimum over all the in-degrees and out-degrees of the vertices of G.

Line Directed Hypergraphs

“…Indeed their underlying digraphs are generalized de Bruijn (resp Kautz) digraphs whose connectivity is known. Therefore using the results of [8,12] we get: As we will see the hyperarc connectivity of the generalized de Bruijn and Kautz dihypergraphs is equal to their minimum degree as soon as s ≥ 2. The result was more difficult to prove.…”

Further Topics in Connectivity

“…D12: [FaFi89,FiFaEs90] For a given digraph G = (V, E) with diameter D, the semigirth, denoted`(G), is the greatest integer`between 1 and D such that for any u, v 2 V , (a) if dist(u, v) <`, the shortest u-v directed walk is unique and there are no u-v directed walks of length dist(u, v) + 1.…”

Super Connectivity of Line Graphs and Digraphs

“…Since then it has been found that many well-known graphs are super connected or super edge-connected. In particular, Soneoka [10] showed that the de Bruijn digraph B(d, n) is super edge-connected for any d ≥ 2 and n ≥ 1; Fábrega and Fiol [5] proved that the Kautz digraph K(d, n) is super edge-connected for any d ≥ 3 and n ≥ 2.…”