Limit cycles and update digraphs in Boolean networks

Boolean networks have been used as models of gene regulation and other biological networks, as well as for other kinds of distributed dynamical systems. One key element in these models is the update schedule, which indicates the order in which states have to be updated. In Salinas (2008) [22] and Aracena et al. (2009) [1], equivalence classes of deterministic update schedules according to the labeled digraph associated to a Boolean network (update digraph) were defined and it was proved that two schedules in the same class yield the same dynamical behavior. In this paper, we study the relations between the update digraphs and the preservation of limit cycles of Boolean networks iterated under non-equivalent update schedules. We show that the related problems lie in the class of NP-hard problems and we prove that the information provided by the update digraphs is not sufficient to determine whether two Boolean networks share limit cycles or not. Besides, we exhibit a polynomial algorithm that works as a necessary condition for two Boolean networks to share limit cycles. Finally, we construct some update schedule classes whose elements share a given limit cycle under certain conditions on the frozen nodes of it.     

Locally-finite connected-homogeneous digraphs

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.     

Powerful digraphs

We introduce the concept of a powerful digraph and establish that a powerful digraph structure is included into the saturated structure of each nonprincipal powerful type p possessing the global pairwise intersection property and the similarity property for the theories of graph structures of type p and some of its first-order definable restrictions (all powerful types in the available theories with finitely many (> 1) pairwise nonisomorphic countable models have this property). We describe the structures of the transitive closures of the saturated powerful digraphs that occur in the models of theories with nonprincipal powerful 1-types provided that the number of nonprincipal 1-types is finite. We prove that a powerful digraph structure, considered in a model of a simple theory, induces an infinite weight, which implies that the powerful digraphs do not occur in the structures of the available classes of the simple theories (like the supersimple or finitely based theories) that do not contain theories with finitely many (> 1) countable models.     

On large vertex-symmetric digraphs

There is special interest in the design of large vertex-symmetric graphs and digraphs as models of interconnection networks for implementing parallelism. In these systems, a large number of nodes are connected with relatively few links and short paths between the nodes, and each node may execute the same communication software without modifications.In this paper, a method for obtaining new general families of large vertex-symmetric digraphs is put forward. To be more precise, from a k-reachable vertex-symmetric digraph and another (k+1)-reachable digraph related to the previous one, and using a new special composition of digraphs, new families of vertex-symmetric digraphs with small diameter are presented. With these families we obtain new vertex-symmetric digraphs that improve various values of the table of the largest known vertex-symmetric (Δ,D)-digraphs. The paper also contains the (Δ,D)-table for vertex-symmetric digraphs, for Δ≤13 and D≤12.     

Highly Arc-Transitive Digraphs With No Homomorphism Onto <Emphasis Type="Bold">Z</Emphasis>

In an infinite digraph D, an edge e' is reachable from an edge e if there exists an alternating walk in D whose initial and terminal edges are e and e'. Reachability is an equivalence relation and if D is 1-arc-transitive, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs. In Combinatorica, 13 (1993), Cameron, Praeger and Wormald asked if there exist highly arc-transitive digraphs (apart from directed cycles) for which the reachability relation is not universal and which do not have a homomorphism onto the two-way infinite directed path (a Cayley digraph of Z with respect to one generator). In view of an earlier result of Praeger in Australas. J. Combin., 3 (1991), such digraphs are either locally infinite or have equal in- and out-degree. In European J. Combin., 18 (1997), Evans gave an affirmative answer by constructing a locally infinite example. For each odd integer n >= 3, a construction of a highly arc-transitive digraph without property Z satisfying the additional properties that its in- and out-degree are equal to 2 and that the reachability equivalence classes induce alternating cycles of length 2n, is given. Furthermore, using the line digraph operator, digraphs having the above properties but with alternating cycles of length 4 are obtained. Received April 12, 1999 Supported in part by "Ministrstvo za šolstvo, znanost in šport Slovenije", research program PO-0506-0101-99.     

On the weak distance-regularity of Moore-type digraphs

We prove that Moore digraphs, and some other classes of extremal digraphs, are weakly distance-regular in the sense that there is an invariance of the number of walks between vertices at a given distance. As weakly distance-regular digraphs, we then compute their complete spectrum from a ‘small’ intersection matrix. This is a very useful tool for deriving some results about their existence and/or their structural properties. For instance, we present here an alternative and unified proof of the existence results on Moore digraphs, Moore bipartite digraphs and, more generally, Moore generalized p-cycles. In addition, we show that the line digraph structure appears as a characteristic property of any Moore generalized p-cycle of diameter D?≥?2p.     

Disjoint quasi‐kernels in digraphs

A quasi‐kernel in a digraph is an independent set of vertices such that any vertex in the digraph can reach some vertex in the set via a directed path of length at most two. Chvátal and Lovász proved that every digraph has a quasi‐kernel. Recently, Gutin et al. raised the question of which digraphs have a pair of disjoint quasi‐kernels. Clearly, a digraph has a pair of disjoint quasi‐kernels cannot contain sinks, that is, vertices of outdegree zero, as each such vertex is necessarily included in a quasi‐kernel. However, there exist digraphs which contain neither sinks nor a pair of disjoint quasi‐kernels. Thus, containing no sinks is not sufficient in general for a digraph to have a pair of disjoint quasi‐kernels. In contrast, we prove that, for several classes of digraphs, the condition of containing no sinks guarantees the existence of a pair of disjoint quasi‐kernels. The classes contain semicomplete multipartite, quasi‐transitive, and locally semicomplete digraphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:251‐260, 2008     

On Vosperian and Superconnected Vertex-Transitive Digraphs

We investigate the structure of a digraph having a transitive automorphism group where every cutset of minimal cardinality consists of all successors or all predecessors of some vertex. We give a complete characterization of vosperian arc-transitive digraphs. It states that an arc-transitive strongly connected digraph is vosperian if and only if it is irreducible. In particular, this is the case if the degree is coprime with the order of the digraph. We give also a complete characterization of vosperian Cayley digraphs and a complete characterization of irreducible superconnected Cayley digraphs. These two last characterizations extend the corresponding ones in Abelian Cayley digraphs and the ones in the undirected case.     

On even factorizations and the chromatic index of the Kautz and de Bruijn digraphs

Motivated by the problem of designing large packet radio networks, we show that the Kautz and de Bruijn digraphs with in- and outdegree d have arc-chromatic index 2d. In order to do this, we introduce the concept of even 1-factorizations. An even 1-factor of a digraph is a spanning subgraph consisting of vertex disjoint loops and even cycles; an even 1-factorization is a partition of the arcs into even 1-factors. We prove that if a digraph admits an even 1-factorization, then so does its line digraph. (In fact, we show that the line digraph admits an even 1-factorization even under a weaker assumption discussed below.) As a consequence, we derive the above property of the Kautz and de Bruijn digraphs relevant to packet radio networks. © 1993 John Wiley & Sons, Inc.     

有向图反能量的一个注记

有向图的反能量是指有向图的反邻接矩阵的能量.本文利用有向图的运算构造出了几类有向图,它们中的每一个都满足有向图的反能量等于其底图的能量.部分回答了Adiga等人在文[The skew energy of a digraph,Linear Algebra Appl.,2010,432:1825-1835]中提出的一个公开问题.     

Revisiting the Comellas-Fiol-Gómez constructions of large digraphs of given degree and diameter

We analyze three constructions of Comellas and Fiol [F. Commellas, M.A. Fiol, Vertex-symmetric digraphs with small diameter, Discrete Applied Mathematics 58 (1995) 1-11] that produce large digraphs of given diameter and degree from smaller starter digraphs. We show that these constructions preserve coverings in the sense that if the starter digraph is a regular lift (in particular, a Cayley digraph), then the resulting digraph has the same property.     

Locally semicomplete digraphs: A generalization of tournaments

In this paper we introduce a new class of directed graphs called locally semicomplete digraphs. These are defined to be those digraphs for which the following holds: for every vertex x the vertices dominated by x induce a semicomplete digraph and the vertices that dominate x induce a semicomplete digraph. (A digraph is semicomplete if for any two distinct vertices u and ν, there is at least one arc between them.) This class contains the class of semicomplete digraphs, but is much more general. In fact, the class of underlying graphs of the locally semi-complete digraphs is precisely the class of proper circular-arc graphs (see [13], Theorem 3). We show that many of the classic theorems for tournaments have natural analogues for locally semicomplete digraphs. For example, every locally semicomplete digraph has a directed Hamiltonian path and every strong locally semicomplete digraph has a Hamiltonian cycle. We also consider connectivity properties, domination orientability, and algorithmic aspects of locally semicomplete digraphs. Some of the results on connectivity are new, even when restricted to semicomplete digraphs.     

Computing transitive closure of bipolar weighted digraphs

We define a bipolar weighted digraph as a weighted digraph together with the sign function on the arcs such that the weight of each arc lies between 0 and 1, and no two parallel arcs have the same sign. Bipolar weighted digraphs are utilized to model so-called fuzzy cognitive maps, which are used in science, engineering, and the social sciences to represent the causal structure of a body of knowledge. It has been noted in the literature that a transitive closure of a bipolar weighted digraph contains useful new information for the fuzzy cognitive map it models.In this paper we ask two questions: what is a sensible and useful definition of transitive closure of a bipolar weighted digraph, and how do we compute it? We give two answers to each of these questions, that is, we present two distinct models. First, we give a review of the fuzzy digraph model, which has been, in a different form and less rigorously, studied previously in the fuzzy systems literature. Second, we carefully develop a probabilistic model, which is related to the notion of network reliability.This paper is intended for a mathematical audience.     

Weakly distance-regular digraphs

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented.     

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.     

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.     

Turán theorems and convexity invariants for directed graphs

This paper is motivated by the desire to evaluate certain classical convexity invariants (specifically, the Helly and Radon numbers) in the context of transitive closure of arcs in the complete digraph. To do so, it is necessary to establish several new Turán type results for digraphs and characterize the associated extremal digraphs. In the case of the Radon number, we establish the following analogue for transitive closure in digraphs of Radon's classical convexity theorem [J. Radon, Mengen konvexer Körper, die einer gemeinsamen Punkt enthalten, Math. Ann. 83 (1921) 113-115]: in a complete digraph on n?7 vertices with >n²/4 arcs, it is possible to partition the arc set into two subsets whose transitive closures have an arc in common.     

Finding cheapest cycles in vertex-weighted quasi-transitive and extended semicomplete digraphs

We consider the problem of finding a minimum cost cycle in a digraph with real-valued costs on the vertices. This problem generalizes the problem of finding a longest cycle and hence is NP-hard for general digraphs. We prove that the problem is solvable in polynomial time for extended semicomplete digraphs and for quasi-transitive digraphs, thereby generalizing a number of previous results on these classes. As a byproduct of our method we develop polynomial algorithms for the following problem: Given a quasi-transitive digraph D with real-valued vertex costs, find, for each j=1,2,…,|V(D)|, j disjoint paths P₁,P₂,…,P_j such that the total cost of these paths is minimum among all collections of j disjoint paths in D.     

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.