Super-connected arc-transitive digraphs

A digraph is said to be super-connected if every minimum vertex cut is the out-neighbor set or in-neighbor set of a vertex. A digraph is said to be reducible, if there are two vertices with the same out-neighbor set or the same in-neighbor set. In this paper, we prove that a strongly connected arc-transitive oriented graph is either reducible or super-connected. Furthermore, if this digraph is also an Abelian Cayley digraph, then it is super-connected.     

Independent sets and non-augmentable paths in arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs

A digraph is arc-locally in-semicomplete if for any pair of adjacent vertices x,y, every in-neighbor of x and every in-neighbor of y either are adjacent or are the same vertex. A digraph is quasi-arc-transitive if for any arc xy, every in-neighbor of x and every out-neighbor of y either are adjacent or are the same vertex. Laborde, Payan and Xuong proposed the following conjecture: Every digraph has an independent set intersecting every non-augmentable path (in particular, every longest path). In this paper, we shall prove that this conjecture is true for arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs.     

A Characterization of Locally Semicomplete CKI-digraphs

A digraph is locally-in semicomplete if for every vertex of D its in-neighborhood induces a semicomplete digraph and it is locally semicomplete if for every vertex of D the in-neighborhood and the out-neighborhood induces a semicomplete digraph. The locally semicomplete digraphs where characterized in 1997 by Bang-Jensen et al. and in 1998 Bang-Jensen and Gutin posed the problem if finding a kernel in a locally-in semicomplete digraph is polynomial or not. A kernel of a digraph is a set of vertices, which is independent and absorbent. A digraph D such that every proper induced subdigraph of D has a kernel is said to be critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. A digraph without an induced CKI-digraph as a subdigraph does have a kernel. We characterize the locally semicomplete digraphs, which are CKI. As a consequence of this characterization we conclude that determinate whether a locally semicomplete digraph is a CKI-digraph or not, is polynomial.     

Sufficient Conditions for Super-Arc-Strongly Connected Oriented Graphs

A digraph D is called super-arc-strongly connected if the arcs of every its minimum arc-disconnected set are incident to or from some vertex in D. A digraph without any directed cycle of length 2 is called an oriented graph. Sufficient conditions for digraphs to be super-arc-strongly connected have been given by several authors. However, closely related conditions for super-arc-strongly connected oriented graphs have little attention until now. In this paper we present some minimum degree and degree sequence conditions for oriented graphs to be super-arc-strongly connected.     

Quasi-transitive digraphs

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interseting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, we show that every strongly connected quasi-transitive digraphs D on at least four vertices has two vertices v₁ and v₂ such that D – v_i is strongly connected for i = 1, 2. A result of tournaments on the existence of a pair of arc-disjoint in- and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore we characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 ≦ k ≦ |V(D)| ? 1 through every vertex and yet they are not hamiltonian. Finally we characterize pancyclic and vertex pancyclic quasi-transitive digraphs. © 1995, John Wiley & Sons, Inc.     

The structure of strong arc-locally in-semicomplete digraphs

A digraph is arc-locally in-semicomplete if for any pair of adjacent vertices $x,y$, every in-neighbor of $x$ and every in-neighbor of $y$ either are adjacent or are the same vertex. In this paper, we study the structure of strong arc-locally in-semicomplete digraphs and prove that a strong arc-locally in-semicomplete digraph is either arc-locally semicomplete or in a special class of digraphs. Using this structural characterization, we show that a 2-strong arc-locally in-semicomplete digraph is arc-locally semicomplete and a conjecture of Bang-Jensen is true.     

Multiplicativity of acyclic local tournaments

A homomorphism of a digraph to another digraph is an edge-preserving vertex mapping. A digraphH is said to be multiplicative if the set of digraphs which do not admit a homomorphism toH is closed under categorical product. In this paper we discuss the multiplicativity of acyclic Hamiltonian digraphs, i.e., acyclic digraphs which contains a Hamiltonian path. As a consequence, we give a complete characterization of acyclic local tournaments with respect to multiplicativity.     

Kernels and some operations in edge-coloured digraphs

Let D be an edge-coloured digraph, V(D) will denote the set of vertices of D; a set N⊆V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following two conditions: For every pair of different vertices u,v∈N there is no monochromatic directed path between them and; for every vertex x∈V(D)−N there is a vertex y∈N such that there is an xy-monochromatic directed path.In this paper we consider some operations on edge-coloured digraphs, and some sufficient conditions for the existence or uniqueness of kernels by monochromatic paths of edge-coloured digraphs formed by these operations from another edge-coloured digraphs.     

On the sum of the total domination numbers of a digraph and its converse

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S has an in-neighbor in S. A dominating set S of D is called a total dominating set of D if the subdigraph induced by S has no isolated vertices. The total domination number of D, denoted by γt(D), is the minimum cardinality of a total dominating set of D. We show that for any connected digraph D of order n≥3, γt(D)+γt(D^? )≤5n/3, where D? is the converse of D. Furthermore, we characterize the oriented trees for which the equality holds.     

Graph Mates

A weighted digraph graph D is said to be doubly stochastic if all the weights of the edges in D are in [0, 1] and sum of the weights of the edges incident to each vertex in D is one. Let Ω(G) be denoted as set of all doubly stochastic digraphs with n vertices. We defined a Graph Mates in Ω(G) and derived a necessary and sufficient condition for two doubly stochastic digraphs are to be a Graph Mates.     

Efficient total domination in digraphs

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X. We study graphs that permit an orientation having such a set and give complexity results and characterizations. Furthermore, we study the computational complexity of the (weighted) efficient total domination problem for several digraph classes. In particular we deal with most of the common generalizations of tournaments, like locally semicomplete and arc-locally semicomplete digraphs.     

Some sufficient conditions for the existence of kernels in infinite digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)−N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of D is an independent set of vertices S of D such that for every z∈V(D)−S for which there exists an (S,z)-arc of D−F, there also exists an (z,S)-arc in D. In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F. As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.     

Connected -tuple twin domination in de Bruijn and Kautz digraphs

For a digraph G, a k-tuple twin dominating set D of G for some fixed k≥1 is a set of vertices such that every vertex is adjacent to at least k vertices in D, and also every vertex is adjacent from at least k vertices in D. If the subgraph of G induced by D is strongly connected, then D is called a connected k-tuple twin dominating set of G. In this paper, we give constructions of minimal connected k-tuple twin dominating sets for de Bruijn digraphs and Kautz digraphs.     

Cycle factors in strongly connected local tournaments

A digraph without loops, multiple arcs and directed cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. A digraph is 2-connected if the removal of an arbitrary vertex results in a strongly connected digraph.In 2004 and 2005, Li and Shu investigated the structure of strongly connected, but not 2-connected tournaments. Using their structural results they were able to give sufficient conditions for a strongly connected tournament T to have complementary cycles or a k-cycle factor, i.e. a set of k vertex disjoint cycles that span the vertex set of T.Inspired by the articles of Li and Shu we develop in this paper the structure necessary for a strongly connected local tournament to be not cycle complementary. Using this structure, we are able to generalize and transfer various results of Li and Shu to the class of local tournaments.     

On the number of connected convex subgraphs of a connected acyclic digraph

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 interest in the area of modern embedded processors technology.     

Descendant-homogeneous digraphs

The descendant setdesc(α) of a vertex α in a digraph D is the set of vertices which can be reached by a directed path from α. A subdigraph of D is finitely generated if it is the union of finitely many descendant sets, and D is descendant-homogeneous if it is vertex transitive and any isomorphism between finitely generated subdigraphs extends to an automorphism. We consider connected descendant-homogeneous digraphs with finite out-valency, specially those which are also highly arc-transitive. We show that these digraphs must be imprimitive. In particular, we study those which can be mapped homomorphically onto Z and show that their descendant sets have only one end.There are examples of descendant-homogeneous digraphs whose descendant sets are rooted trees. We show that these are highly arc-transitive and do not admit a homomorphism onto Z. The first example (Evans (1997) [6]) known to the authors of a descendant-homogeneous digraph (which led us to formulate the definition) is of this type. We construct infinitely many other descendant-homogeneous digraphs, and also uncountably many digraphs whose descendant sets are rooted trees but which are descendant-homogeneous only in a weaker sense, and give a number of other examples.     

On the number of noncritical vertices in strongly connected digraphs

By definition, a vertex w of a strongly connected (or, simply, strong) digraph D is noncritical if the subgraph D — w is also strongly connected. We prove that if the minimal out (or in) degree k of D is at least 2, then there are at least k noncritical vertices in D. In contrast to the case of undirected graphs, this bound cannot be sharpened, for a given k, even for digraphs of large order. Moreover, we show that if the valency of any vertex of a strong digraph of order n is at least 3/4n, then it contains at least two noncritical vertices. The proof makes use of the results of the theory of maximal proper strong subgraphs established by Mader and developed by the present author. We also construct a counterpart of this theory for biconnected (undirected) graphs.     

On Hamiltonian Powers of Digraphs

 For a strongly connected digraph D, the k-th power D ^k of D is the digraph with the same set of vertices, a vertex x being joined to a vertex y in D ^k if the directed distance from x to y in D is less than or equal to k. It follows from a result of Ghouila-Houri that for every digraph D on n vertices and for every k≥n/2, D ^k is hamiltonian. In the paper we characterize these digraphs D of odd order whose (⌈n/2 ⌉−1)-th power is hamiltonian. Revised: June 13, 1997     

On the number of cycles in local tournaments

A digraph without loops, multiple arcs and directed cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. A vertex of a strongly connected digraph is called a non-separating vertex if its removal preserves the strong connectivity of the digraph in question.In 1990, Bang-Jensen showed that a strongly connected local tournament does not have any non-separating vertices if and only if it is a directed cycle. Guo and Volkmann extended this result in 1994. They determined the strongly connected local tournament with exactly one non-separating vertex. In the first part of this paper we characterize the class of strongly connected local tournaments with exactly two non-separating vertices.In the second part of the paper we consider the following problem: Given a strongly connected local tournament D of order n with at least n+2 arcs and an integer 3≤r≤n. How many directed cycles of length r exist in D? For tournaments this problem was treated by Moon in 1966 and Las Vergnas in 1975. A reformulation of the results of the first part shows that we have characterized the class of strongly connected local tournaments with exactly two directed cycles of length n−1. Among other things we show that D has at least n−r+1 directed cycles of length r for 4≤r≤n−1 unless it has a special structure. Moreover, we characterize the class of local tournaments with exactly n−r+1 directed cycles of length r for 4≤r≤n−1 which generalizes a result of Las Vergnas.