The Graph Drawing community uses test suites for comparing layout quality and e ciency. Those suites often claim to collect randomly generated graphs, although in most cases randomness is a loosely de ned notion. We propose a simple algorithm for generating acyclic digraphs with a given number of vertices uniformly at random. Applying standard combinatorial techniques, we describe the overall s...

In this note we show that the edge-connectivity λ(G × H) of the direct product of graphs G and H is bounded below by min{λ(G)|E(H)|, λ(H)|E(G)|, δ(G × H)} and above by min{2λ(G)|E(H)|, 2λ(H)|E(G)|, δ(G×H)} except in some special cases when G is a relatively small bipartite graph, or both graphs are bipartite. Several upper bounds on the vertex-connectivity of the direct product of graphs are al...

A digraph D is connected if the underlying undirected graph of D is connected. A subgraph H of an acyclic digraph D is convex if there is no directed path between vertices of H which contains an arc not in H. We find the minimum and maximum possible number of connected convex subgraphs in a connected acyclic digraph of order n. Connected convex subgraphs of connected acyclic digraphs are of int...

We consider the relationships between Cayley digraphs and their coset graphs with respect to subgroups and obtain some general results on homomorphism and broadcasting between them. We also derive a general factorization theorem on subgraphs of Cayley digraphs by their automorphism groups. We discuss the applications of these results to well-known interconnection networks such as the butterfly ...

Let H be a graph with a designated vertex s, where edges are weighted by nonnegative reals. Splitting edges e = {u, s} and e′ = {s, v} at s is an operation that reduces the weight of each of e and e′ by a real δ > 0 while increasing the weight of edge {u, v} by δ. It is known that all edges incident to s can be split off while preserving the edge-connectivity of H and that such a complete split...

In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs G1 and G2 are equal to κ1v2 and min{λ1v 2 2 , δ2 +δ1v2}, respectively, where δi, κi, λi and vi denote the minimum degree, the connectivity, the edge-connectivity and the number of vertices of Gi, respectively. We also obtain that the edge-connectivity of the direct product of K2 an...

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum out-degrees (d1; d2) of its partite sets of vertices. It has been proved that, when d1d2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist w...

The modular decomposition of a graph is a canonical representation of its modules. Algorithms for computing the modular decomposition of directed and undirected graphs differ significantly, with the undirected case being simpler, and algorithms for directed graphs often work by reducing the problem to decomposing undirected graphs. In this paper we show that transitive acyclic digraphs have the...

In this article we generalize the concept of line digraphs to line dihypergraphs. We give some general properties in particular concerning connectivity parameters of dihypergraphs and their line dihypergraphs, like the fact that the arc connectivity of a line dihypergraph is greater than or equal to that of the original dihypergraph. Then we show that the De Bruijn and Kautz dihypergraphs (whic...