Transitive Digraphs Research Articles

Weak ψ-Contractions on Directed Graphs with Applications to Integral Equations

This article deals with a few outcomes ensuring the fixed points of a weak (G,ψ)-contraction map of metric spaces comprised with a reflexive and transitive digraph G. To validate our findings, we furnish several examples. The findings we obtain enable us to seek out the unique solution of a nonlinear integral equation. The outcomes presented herewith sharpen, subsume, unify, improve, enrich, and compile a number of existing theorems.

Nonlinear Contractions Employing Digraphs and Comparison Functions with an Application to Singular Fractional Differential Equations

After the initiation of Jachymski’s contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem.

Arc-disjoint out- and in-branchings in compositions of digraphs

An out-branching Bu+ (in-branching Bu−) in a digraph D is a connected spanning subdigraph of D in which every vertex except the vertex u, called the root, has in-degree (out-degree) one. A good(u,v)-pair in D is a pair of branchings Bu+,Bv− which have no arc in common. Thomassen proved that it is NP-complete to decide if a digraph has any good pair. A digraph is semicomplete if it has no pair of non-adjacent vertices. A semicomplete composition is any digraph D which is obtained from a semicomplete digraph S by substituting an arbitrary digraph Hx for each vertex x of S.Recently the authors of this paper gave a complete classification of semicomplete digraphs which have a good (u,v)-pair, where u,v are prescribed vertices. They also gave a polynomial algorithm which for a given semicomplete digraph D and vertices u,v of D, either produces a good (u,v)-pair in D or a certificate that D has no such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of characterizing semicomplete compositions which have a good (u,v)-pair for given vertices u,v. Our solution implies that the problem of deciding the existence of a good (u,v)-pair and finding such a pair when it exists is polynomially solvable for all semicomplete compositions. We also completely solve the problem of deciding the existence of a good (u,v)-pair and finding one when it exists for digraphs that are compositions of transitive digraphs. Combining these two results we obtain a polynomial algorithm for deciding whether a given quasi-transitive digraph D has a good (u,v)-pair for given vertices u,v of D. This proves a conjecture of Bang-Jensen and Gutin from 1998.

Witten Complex of Transitive Digraph and Its Convergence

Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.

Action graph of a semigroup act & its functorial connection

In this paper we define C-induced action graph G(S,a,C;A) corresponding to a semigroup act (S,a,A) and a subset C of S. This generalizes many interesting graphs including Cayley Graph of groups and semigroups, Transformation Graphs (TRAG), Group Action Graphs (GAG), Derangement Action Graphs, Directed Power Graphs of Semigroups etc. We focus on the case when C = S and name the digraph, so obtained, as Action Graph of a Semigroup Act (S, a, A). Some basic structural properties of this graph follow from algebraic properties of the underlying semigroup and its action on the set. Action graph of a strongly faithful act is also studied and graph theoretic characterization of a strongly faithful semigroup act as well as that of idempotents in a semigroup are obtained. We introduce the notion of strongly transitive digraphs and based on this we characterize action graphs of semigroup acts in the class of simple digraphs. The simple fact that morphism between semigroup acts leads to digraph homomorphism between corresponding action graphs, motivates us to represent action graph construction as a functor from the category of semigroup acts to the category of certain digraphs. We capture its functorial properties, some of which signify previous results in terms of Category Theory.

Homology of transitive digraphs and discrete spaces

We prove that for transitive digraphs path homology, and therefore also Alexandroff homology, coincides with singular cubic homology. Also, discrete topological spaces are defined that are natural analogues of standard topological cubes. Using them, the singular cubic homology of discrete topological spaces is defined, and it is proved that these homology groups coincide with the Alexandroff homology groups. Bibliography: 24 titles.

Universe as a Graph (Ramsey Approach to Analysis of Physical Systems)

Ramsey theory is implemented to the analysis of physical systems. Physical interactions may be very generally classified as attractive and repulsive. This classification creates the premises for the application of the Ramsey theory to the analysis of physical systems. The basic notions of mathematical logic, such as transitivity and intransitivity, become key ones for understanding of physical systems. The Ramsey theory explains why nature prefers cubic lattices over hexagonal ones for systems built of electric or magnetic dipoles. The Ramsey approach may be applied to the analysis of mechanical systems when actual and virtual paths between the states in the configurational space are considered. Irreversible mechanical and thermodynamic processes are seen as directed graphs (digraphs). Chains of irreversible processes appear as transitive tournaments. These digraphs are acyclic; the transitive digraphs necessarily contain the Hamiltonian (or traceable) path. The set of states in the phase space of the physical system, between which irreversible processes are possible, is considered. The Hamiltonian path of the digraph arising from the graph uniting these states is a relativistic invariant. Applications of the Ramsey theory to the general relativity are possible. Reconsideration of the concept of “simultaneity” within the Ramsey approach is reported.

Homologies of transitive digraphs and discrete spaces

В работе доказано, что для транзитивных орграфов гомологии путей и, следовательно, гомологии Александрова совпадают с сингулярными кубическими гомологиями. Также в работе определены дискретные топологические пространства, являющиеся естественными аналогами стандартных топологических кубов. С их помощью определены сингулярные кубические гомологии дискретных топологических пространств и доказано, что эти группы гомологий совпадают с гомологиями Александрова. Библиография: 24 названия.

On enumeration of labeled connected transitive digraphs

If Tc0(n) is the number of all labeled connected transitive digraphs defined on a set of n elements, then the formula Tc0(n) = S pˉ = (p1, ..., pk)| = n (-1)n-k (np) W c (pˉ) is true. The summation is over all sequences pˉ= (p1, ..., pk) ∈ ℕk1 such that p1 +...+ pk = n. Term W c (pˉ) denotes the number of labeled connected transitive digraphs of a special form, depending on the parameter pˉ.

Fuzzy relational Galois connections between fuzzy transitive digraphs

We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set.

Pinning Asymptotic Stabilization of Probabilistic Boolean Networks: A Digraph Approach

This article investigates the pinning asymptotic stabilization of probabilistic Boolean networks (PBNs) by a digraph approach. In order to stabilize the PBN asymptotically, a mode-independent pinning control (MIPC) is designed to make the target state a fixed point, and transform the state transition digraph into one that has a spanning branching rooted at the target vertex. It is shown that if there is a mode-dependent pinning control that can asymptotically stabilize the PBN, then there must exist an MIPC that can do the same with fewer pinned nodes and control inputs. A necessary and sufficient condition is given to verify the feasibility of a set of pinned nodes based on an auxiliary digraph. Three algorithms are proposed to find a feasible set of pinned nodes with the minimum cardinality. The main results are extended to the case where the target is a limit cycle. Compared with the existing results, the total computational complexity of these algorithms is strongly reduced. The obtained results are applied to a numerical example and a cell apoptosis network.

Distance–transitive digraphs: Descendant-homogeneity, property [formula omitted] and reachability

A digraph is distance–transitive if for every integer s≥0, the automorphism group of the digraph is transitive on pairs (u,v) of vertices for which there is a directed path of length s from u to v, but no directed path of length t<s. This implies vertex and edge transitivity but is weaker than being highly-arc-transitive. A digraph is said to have property Z if it has a homomorphism onto the two-way infinite directed path. In a 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 on the edge set of D and if D is transitive on edges, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs.We investigate the class of infinite distance–transitive digraphs of finite out-valency. First, we show that earlier results, proved in the context of highly-arc-transitive digraphs, hold for the class of distance–transitive digraphs. Second, we show that if D is a weakly descendant–homogeneous digraph in our class then either (1) D has property Z and the reachability relation is not universal; or (2) D does not have property Z, the reachability relation is universal and D has infinite in-valency. Finally, we describe some distance–transitive, weakly descendant–homogeneous digraphs for which the reachability relation is not universal.

On some formulas in the problem of enumeration of finite labeled topologies

Among the unsolved problems of enumeration of graphs, one of the most difficult is the problem of enumeration of transitive digraphs, which is equivalent to the problem of enumeration of finite partial orders and is equivalent to the problem of enumeration of finite labeled [Formula: see text]-topologies. The sequence [Formula: see text], where [Formula: see text] is the number of all labeled [Formula: see text]-topologies defined on an n-set, has been studied from different points of view. In the previous work of the second author, a formula was obtained in which the number [Formula: see text] is represented as a linear combination of numbers [Formula: see text] (where the sequences [Formula: see text] are compositions of the number n). Recurrent relations between individual numbers [Formula: see text] were obtained. In this paper, new recursive formulas between separate numbers [Formula: see text] are obtained.

$(k,H)$-kernels in nearly tournaments

Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $\rho : A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0, \ldots , n-1 \}$, we say that there is an obstruction on $x_{i}$ whenever $(\rho(x_{i-1}, x_{i}), \rho (x_{i}, x_{i+1})) \notin A(H)$ (when $x_{0} = x_{n}$ the indices are taken modulo $n$). We denote by $O_{H}(W)$ the set $\{ i \in \{0, \ldots , n-1 \} :$ there is an obstruction on $x_{i} \}$. The $H$-length of $W$, denoted by $l_{H}(W)$, is defined by $|O_{H}(W)|+1$ whenever $x_{0} \neq x_{n}$, or $|O_{H}(W)|$ in other case. A $(k, H)$-kernel of an $H$-colored digraph $D$ ($k \geq 2$) is a subset of vertices of $D$, say $S$, such that, for every pair of different vertices in $S$, every path between them has $H$-length at least $k$, and for every vertex $x \in V(D) \setminus S$ there exists an $xS$-path with $H$-length at most $k-1$. This concept widely generalize previous nice concepts as kernel, $k$-kernel, kernel by monochromatic paths, kernel by properly colored paths, and $H$-kernel. In this paper, we will study the existence of $(k,H)$-kernels in interesting classes of digraphs, called nearly tournaments, which have been large and widely studied due its applications and theoretical results. We will show several conditions that guarantee the existence of $(k,H)$-kernel in tournaments, $r$-transitive digraphs, $r$-quasi-transitive digraphs, multipartite tournaments, and local tournaments.

Efficient computation of the oriented chromatic number of recursively defined digraphs

In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph G=(V,E), an oriented r-coloring for G is a partition of the vertex set V into r independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of G is the smallest integer r such that G permits an oriented r-coloring. Deciding whether an acyclic digraph has an oriented 4-coloring is NP-hard, which motivates to consider the problem on special graph classes.In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 7 for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure.

Transitive digraph groups

In the present paper the class of digraph groups is introduced by analogy with the prominent class of “graph groups”. Digraph groups implicitly present in the previous work by the author on the automorphism groups of graph groups. The structure of transitive digraph groups is described in terms of the Levi decomposition into the semidirect product of the Steinberg groups and the nilpotent radical.

Relational Galois connections between transitive digraphs: Characterization and construction

This paper focuses on a twofold relational generalization of the notion of Galois connection. It is twofold because it is defined between sets endowed with arbitrary transitive relations and, moreover, both components of the connection are relations, not necessarily functions. A characterization theorem of the notion of relational Galois connection is provided and, then, it is proved that a suitable notion of closure can be obtained within this framework. Finally, we state a necessary and sufficient condition that allows to build a relational Galois connection starting from a single transitive digraph and a single binary relation.

Almost Complete and Equable Heteroclinic Networks

Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the relation between the heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although robust realizations are typically not complete (i.e. not all unstable manifolds of nodes are part of the network), they can be almost complete (i.e. complete up to a set of zero measure within the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). We show there are almost complete and equable realizations that can be closed by adding a number of extra nodes and connections. We discuss some examples and describe a sense in which an equable almost complete network embedding is an optimal description of stochastically perturbed motion on the network.

The Outputs Robustness of Boolean Control Networks via Pinning Control

The outputs robustness is a property where the outputs of a system are insensitive to disturbances, and the property plays an important role in biological systems or engineering design. This paper presents an approach to analyze the outputs robustness with respect to disturbances (w.r.t. disturbances) for Boolean control networks (BCNs). Based on the wiring digraph of a BCN rather than the state transition digraph, an algorithm is proposed to construct a corresponding permutation digraph and permutation system. Then, we prove that if there exists a pinning controller such that the outputs of a permutation system are robust w.r.t. disturbances, then there must also exist another pinning controller such that the outputs of the corresponding original systems achieve robustness. Furthermore, pinning controllers are designed based on the neighbors of the pinned-nodes rather than all of the nodes. Finally, the proposed method is well demonstrated by a reduced signal transduction network.

Unions of rank/trace complete preorders

A zero pattern algebra is a matrix algebra determined by a pattern of zeros corresponding to a preorder (i.e., a reflexive transitive digraph). The issue studied in [6] is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A-valued functions. As has been established in [6], there is a necessary condition involving certain rank/trace requirements. Algebras for which this necessary condition is also sufficient are said to be rank/trace complete. Several classes of rank/trace complete algebras are identified in [6]. Also operations are described there for producing rank/trace complete algebras out of given ones. A basic operation of that type is taking a disjoint union. In the present article this simple operation is generalized to a more involved one which preserves rank/trace completeness too. The new operation turns out to be a useful tool for establishing rank/trace completeness in situations that up to now could not be handled. One of the positive results is concerned with bouquets, a special type of rooted trees. It provides a partial answer to an issue raised in [7].