Thomassen proved that there is no degree of strong connectivity which guarantees a cycle through two given vertices in a digraph (Combinatorica 11 (1991) 393-395). In this paper we consider a large family of digraphs, including symmetric digraphs (i.e. digraphs obtained from undirected graphs by replacing each edge by a directed cycle of length two), semicomplete bipartite digraphs, locally sem...

The linear ordering problem consists of finding an acyclic tournament in a complete weighted digraph of maximum weight. It is one of the classical NP-hard combinatorial optimization problems. This paper surveys a collection of heuristics and metaheuristic algorithms for finding near-optimal solutions and reports about extensive computational experiments with them. We also present the new benchm...

Let $D$ be a digraph, let $p \geq 1$ an integer, and $f: V(D) \to \mathbb{N}_0^p$ vector function with $f=(f_1,f_2,\ldots,f_p)$. We say that has $f$-partition if there is partition $(V_1,V_2,\ldots,V_p)$ of the vertex set such that, for all $i \in [1,p]$, digraph $D_i=D[V_i]$ weakly $f_i$-degenerate, is, in every nonempty subdigraph $D'$ $D_i$ $v$ $\min\{d_{D'}^+(v), d_{D'}^-(v)\} < f_i(v)$. In...

Seymour’s second neighbourhood conjecture asserts that every oriented graph has a vertex whose second out-neighbourhood is at least as large as its out-neighbourhood. In this paper, we prove that the conjecture holds for quasi-transitive oriented graphs, which is a superclass of tournaments and transitive acyclic digraphs. A digraph D is called quasitransitive is for every pair xy, yz of arcs b...

We present a parallel algorithm for the problem of computing the transitive closure for an acyclic digraph D with n vertices and m edges. We use the BSP/CGM model of parallel computing. Our algorithm uses O(log p) rounds of communications with p processors, where p n, and each processor has O(mn p ) local memory. The local computation of each processor is equal to the product of the number of e...

Given a weighted digraph D, finding the longest simple path is well known to be NPhard. Furthermore, even giving an approximation algorithm is known to be NP-hard. In this paper we describe an efficient heuristic algorithm for finding long simple paths, using an hybrid approach of DFS and pseudo-topological orders, a a generalization of topological orders to non acyclic graphs, via a process we...

If D = (V,A) is a digraph, its niche hypergraph NH(D) = (V, E) has the edge set E = {e ⊆ V | |e| ≥ 2 ∧ ∃v ∈ V : e = N− D (v) ∨ e = N + D (v)}. Niche hypergraphs generalize the well-known niche graphs (cf. [?]) and are closely related to competition hypergraphs (cf. [?]) as well as double competition hypergraphs (cf. [?]). We present several properties of niche hypergraphs of acyclic digraphs.

A new topological sorting algorithm is formulated using the parallel computation approach. The time complexity of this algorithm is of the order of the longest distance between a source node and a sink node in an acyclic digraph representing the partial orderings between elements. An implementation of this algorithm with an SIMD machine is discussed. To avoid contention for logical resources, a...

We give an algorithm with complexity O(f(R) 2 kn) for the integer multiflow problem on instances (G,H, r, c) with G an acyclic planar digraph and r + c Eulerian. Here, f is a polynomial function, n = |V (G)|, k = |E(H)| and R is the maximum request maxh∈E(H) r(h). When k is fixed, this gives a polynomial algorithm for the arc-disjoint paths problem under the same hypothesis.

A set of vertices of a graph whose removal leaves an acyclic graph is called a decycling set of the graph. The minimum size of a decycling set of a graph G is referred to as the decycling number ofG. Let f(d, n) be the decycling number of the generalized Kautz digraph GK(d, n). In this paper, we obtain the upper bound of f(d, n) for all n ≥ d ≥ 2.