We prove that the maximal number of directed edges in a vertex-critical strongly connected simple digraph on n vertices is (

In this paper, we deal with the following decision problem: given a conjunctive Boolean network defined by its interaction digraph, does it have limit cycle of length k? We prove that problem is NP-complete in general if k parameter and P digraph strongly connected. The case where fixed, but not connected, remains open. Furthermore, study some variations network, there exist block-sequential (r...

In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of those cuts. We give a conditional lower bound that provides evidence that our algorithms may be tight up to a sub-polynomial factors. As an additional resul...

A strongly connected digraph is called a cactoid-type if each of its blocks consisting finitely many oriented cycles sharing common directed path. In this article, we find...

4 We prove that a number of natural problems concerning the existence of arc-disjoint directed 5 and “undirected” (spanning) subdigraphs in a digraph are NP-complete. Among these are the 6 following of which the first settles an open problem due to Thomassé (see e.g. [1, Problem 9.9.7] 7 and [3, 5]) and the second settles an open problem posed in [3]. 8 • Given a directed graph D and a vertex s...

This article will generally follow the notation and terminology defined in [1]. A digraph is called strongly connected or strong if for every pari of vertices u and v there exists a directed path from u to v and a directed path from v to u. Let k be a positive integer. A digraph G is k-connected if the removal of any set of fewer than k vertices results in a strong digraph. A tournament with n ...

It has been proved that sphericity testing for digraphs is an NP-complete problem. Here, we investigate sphericity of 3-connected single source digraphs. We provide a new combinatorial characterization of sphericity and give a linear time algorithm for sphericity testing. Our algorithm tests whether a 3-connected single source digraph with $n$ vertices is spherical in $O(n)$ time.

Given a matrix M of size n, the digraph D on n vertices is said to be the digraph of M , when Mij 6= 0 if and only if (vi, vj) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, − → L D, is the digraph of a unitary matrix if and only if D is Eulerian. It...

Bollobás and Scott proved that if the weighted outdegree of every vertex of a weighted digraph is at least 1, then it contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result is an extension of a corresponding theorem of Bondy and Fan in weighted graphs. Besides, we give examples showing the invalidity of genera...

For given nonnegative integers k, s an upper bound on the minimum number of vertices of a strongly connected digraph with exactly k kernels and s solutions is presented.