Acyclic digraphs arise in many natural and artificial processes. Among the broader set, dynamic citation networks represent an important type of acyclic digraph. For example, the study of such networks includes the spread of ideas through academic citations, the spread of innovation through patent citations, and the development of precedent in common law systems. The specific dynamics that prod...

A deterministic finite automaton (dfa) of n states over a k-letter alphabet can be seen as a digraph with n vertices which all have exactly k labeled out-arcs (k-out digraph). In 1973 Grusho [16] first proved that with high probability (whp) in a random k-out digraph there is a strongly connected component (scc) of linear size that is reachable from all vertices, i.e., a giant. He also proved t...

We consider directed graphs without loops and multiple edges, where the exclusion of multiple edges means that two vertices cannot be joined by two edges of the same orientation. Let L I ,. . ., LQ be given digraphs. What is the maximum number of edges a digraph can have if it does not contain any L i as a subgraph and has given number of vertices? We shall prove the existence of a sequence of ...

The linear ordering polytope P~.o is defined as the convex hull of all the incidence vectors of the acyclic tournaments on n nodes, it is known that for every facet of P~.o there corresponds a digraph inducing it. Let D be a digraph that induces a facet-defining inequality for P~.o, that is nonequivalent to a trivial inequality or to a 3-dicycle inequality. We show that for such a digraph the f...

In this article, a q-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in random acyclic digraph studied Simon, Crippa, Collenberg 1993. As one corollaries our main theorem, we get results Dilcher Andrews, Simon. theorem involves surprising new divisor function σs(n), which denote σs,z(n). Analytic properties σs,z(n) are also studied...

In 1985, Mader conjectured that for every acyclic digraph F there exists K = K(F) such D with minimum out-degree at least contains a subdivision of F. This conjecture remains widely open, even digraphs on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases Mader’s problem made the following conjecture: ℓ ≥ 2 K(ℓ) orientation cycle length ℓ. We prove...

We developed, and computer-checked by means of the Ref verifier, a formal proof that every weakly extensional, acyclic (finite) digraph can be decorated injectively à la Mostowski by finite sets so that its arcs mimic membership. We managed to have one sink decorated with ∅ by this injection. We likewise proved that a graph whatsoever admits a weakly extensional and acyclic orientation; consequ...

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...

Hefner, K.A.S., K.F. Jones, S. Kim, J.R. Lundgren and F.S. Roberts, (i,j) competition graphs, Discrete Applied Mathematics 32 (1991) 241-262. If D is an acyclic digraph, its competition graph has the same vertex set as D and an edge between vertices x and y if and only if for some vertex u, there are arcs (_q u) and (_Y, u) in D. We study competition graphs of acyclic digraphs D when the indegr...

Let f(k) be the smallest integer such that every f(k)-chromatic digraph contains every oriented tree of order k. Burr proved f(k) ≤ (k − 1)2 in general, and conjectured f(k) = 2k − 2. Burr also proved that every (8k − 9)-chromatic digraph contains every antidirected tree. We improve both of Burr’s bounds. We show that f(k) ≤ k2/2 − k/2 + 1 and that every antidirected tree of order k is containe...