An out-branching and an in-branching of a digraph D are called kdistinct if each of them has k arcs absent in the other. Bang-Jensen, Saurabh and Simonsen (2016) proved that the problem of deciding whether a strongly connected digraph D has k-distinct out-branching and inbranching is fixed-parameter tractable (FPT) when parameterized by k. They asked whether the problem remains FPT when extende...

If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is called almost regular. Recently, L. Volkmann and S. Winzen showed that ev...

A digraph without loops, multiple arcs and directed cycles of length two is called an in-tournament if the set of in-neighbors of every vertex induces a tournament. A local tournament is an in-tournament such that the set of out-neighbors of every vertex induces a tournament as well. Let p ≥ 2 be an integer and let T be a strongly connected tournament such that every vertex has at least p posit...

A digraph is a symmetric cycle if it is symmetric and its underlying graph is a cycle. It is proved that if D is an asymmetric digraph not containing a symmetric cycle, then D remains asymmetric after removing some vertex. It is also showed that each digraph D without a symmetric cycle, whose underlying graph is connected, contains a vertex which is a common fixed point of all automorphisms of D.

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by CO(D) (CC(D)) and its size by co(D) (cc(D)). Gutin, Johnstone...

This article studies the cluster synchronization problem of coupled nonlinear systems with directed topology and competitive relationships. We assume that nodes within same have intrinsic dynamics, while dynamics belonging to different clusters are different. In cluster, there only exist cooperative relationships, may relationships among clusters. Under assumptions dynamic each node satisfies o...

We study the maximum number of 2xed points of boolean networks with local update function AND–OR. We prove that this number for networks with connected digraph is 2(n−1)=2 for n odd and 2(n−2)=2 + 1 for n even if the digraph has not loops; and 2n−1 + 1 otherwise, where n is the number of nodes of the digraph. We also exhibit some networks reaching these bounds. To obtain these results we constr...

Tarjan [9], describes how depth first search can be used to identify Strongly Connected Components (SCC) of a directed graph in linear time. It is standard to study Tarjan’s SCC algorithm in most senior undergraduate or introductory graduate computer science algorithms courses. In this paper we introduce the concept of a twinless strongly connected component (TSCC) of a directed graph. Loosely ...