Simple Digraph Research Articles

A new parallel algorithm for vertex priorities of data flow acyclic digraphs

Data flow acyclic directed graphs (digraph) are widely used to describe the data dependency of mesh-based scientific computing. The parallel execution of such digraphs can approximately depict the flowchart of parallel computing. During the period of parallel execution, vertex priorities are key performance factors. This paper firstly takes the distributed digraph and its resource-constrained parallel scheduling as the vertex priorities model, and then presents a new parallel algorithm for the solution of vertex priorities using the well-known technique of forward---backward iterations. Especially, in each iteration, a more efficient vertex ranking strategy is proposed. In the case of simple digraphs, both theoretical analysis and benchmarks show that the vertex priorities produced by such an algorithm will make the digraph scheduling time converge non-increasingly with the number of iterations. In other cases of non-simple digraphs, benchmarks also show that the new algorithm is superior to many traditional approaches. Embedding the new algorithm into the heuristic framework for the parallel sweeping solution of neutron transport applications, the new vertex priorities improve the performance by 20 % or so while the number of processors scales up from 32 to 2048.

Signed Roman domination in digraphs

Let $$D$$D be a finite and simple digraph with vertex set $$V(D)$$V(D) and arc set $$A(D)$$A(D). A signed Roman dominating function (SRDF) on the digraph $$D$$D is a function $$f:V(D)\rightarrow \{-1,1,2\}$$f:V(D)?{-1,1,2} satisfying the conditions that (i) $$\sum _{x\in N^-[v]}f(x)\ge 1$$?x?N-[v]f(x)?1 for each $$v\in V(D)$$v?V(D), where $$N^-[v]$$N-[v] consists of $$v$$v and all in-neighbors of $$v$$v, and (ii) every vertex $$u$$u for which $$f(u)=-1$$f(u)=-1 has an in-neighbor $$v$$v for which $$f(v)=2$$f(v)=2. The weight of an SRDF $$f$$f is $$w(f)=\sum _{v\in V(D)}f(v)$$w(f)=?v?V(D)f(v). The signed Roman domination number $$\gamma _{sR}(D)$$?sR(D) of $$D$$D is the minimum weight of an SRDF on $$D$$D. In this paper we initiate the study of the signed Roman domination number of digraphs, and we present different bounds on $$\gamma _{sR}(D)$$?sR(D). In addition, we determine the signed Roman domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the signed Roman domination number $$\gamma _{sR}(G)$$?sR(G) of graphs $$G$$G.

New skew Laplacian energy of simple digraphs

For a simple digraph G of order n with vertex set {v1, v2, . . . , vn}, let d+i and d − i denote the out-degree and in-degree of a vertex vi in G, respectively. Let D (G) = diag(d+1 , d + 2 , . . . , d + n ) and D−(G) = diag(d1 , d − 2 , . . . , d − n ). In this paper we introduce SL(G) = D(G)−S(G) to be a new kind of skew Laplacian matrix of G, where D(G) = D+(G)−D−(G) and S(G) is the skew-adjacency matrix of G, and from which we define the skew Laplacian energy SLE(G) of G as the sum of the norms of all the eigenvalues of SL(G). Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.

Minimum feedback arc set of m-free digraphs

For a simple digraph G, let β(G) be the size of the smallest subset X⊆E(G) such that G−X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called m-free if G has no directed cycles of length at most m. This paper proves that β(G)⩽1m−2γ(G) for any m-free digraph G, which generalizes some known results.

Minimum rank, maximum nullity, and zero forcing number of simple digraphs

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simple digraphs are established. The effect of deletion of a vertex on minimum rank or zero forcing number is analyzed, and simple digraphs having very low or very high zero forcing number are characterized.

On Sullivan’s conjecture on cycles in 4-free and 5-free digraphs

For a simple digraph G, let β(G) be the size of the smallest subset X ⊆ E(G) such that G−X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β(G) ≤ 0.3819γ(G) if G is a 4-free digraph, and β(G) ≤ 0.2679γ(G) if G is a 5-free digraph. These improve the results of Sullivan in 2008.

Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction–diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.

Immersing complete digraphs

We consider the problem of immersing the complete digraph on t vertices in a simple digraph. For Eulerian digraphs, we show that such an immersion always exists whenever the minimum degree is at least t(t−1), and, for t≤4, minimum degree at least t−1 suffices. On the other hand, we show that there exist non-Eulerian digraphs with all vertices of arbitrarily high indegree and outdegree which do not contain an immersion of the complete digraph on three vertices. As a side result, we obtain a construction of digraphs with large outdegree in which all cycles have odd length, simplifying a former construction of such graphs by Thomassen.

Constructing and sampling directed graphs with given degree sequences

The interactions between the components of complex networks are often directed. Proper modeling of such systems frequently requires the construction of ensembles of digraphs with a given sequence of in- and out-degrees. As the number of simple labeled graphs with a given degree sequence is typically very large even for short sequences, sampling methods are needed for statistical studies. Currently, there are two main classes of methods that generate samples. One of the existing methods first generates a restricted class of graphs and then uses a Markov chain Monte-Carlo algorithm based on edge swaps to generate other realizations. As the mixing time of this process is still unknown, the independence of the samples is not well controlled. The other class of methods is based on the configuration model that may lead to unacceptably many sample rejections due to self-loops and multiple edges. Here we present an algorithm that can directly construct all possible realizations of a given bi-degree sequence by simple digraphs. Our method is rejection-free, guarantees the independence of the constructed samples and provides their weight. The weights can then be used to compute statistical averages of network observables as if they were obtained from uniformly distributed sampling or from any other chosen distribution.

Oscillations in non-mass action kinetics models of biochemical reaction networks arising from pairs of subnetworks

It is well known that oscillations in models of biochemical reaction networks can arise as a result of a single negative cycle. On the other hand, methods for finding general network conditions for potential oscillations in large biochemical reaction networks containing many cycles are not well developed. A biochemical reaction network with any number of species is represented by a simple digraph and is modeled by an ordinary differential equation (ODE) system with non-mass action kinetics. The obtained graph-theoretic condition generalizes the negative cycle condition for oscillations in ODE models to the existence of a pair of subnetworks, where each subnetwork contains an even number of positive cycles. The technique is illustrated with a model of genetic regulation.

Signed 2-independence in digraphs

Let D be a finite and simple digraph with vertex set V ( D ) , and let f : V ( D ) → { − 1 , 1 } be a two-valued function. If ∑ x ∈ N − [ v ] f ( x ) ≤ 1 for each v ∈ V ( D ) , where N − [ v ] consists of v and all vertices of D from which arcs go into v , then f is a signed 2-independence function on D . The sum f ( V ( D ) ) is called the weight w ( f ) of f . The maximum of weights w ( f ) , taken over all signed 2-independence functions f on D , is the signed 2-independence number α s 2 ( D ) of D . In this work, we mainly present upper bounds on α s 2 ( D ) , as for example α s 2 ( D ) ≤ n − 2 ⌈ Δ − / 2 ⌉ and α s 2 ( D ) ≤ Δ + + 1 − 2 ⌈ δ − 2 ⌉ Δ + + 1 ⋅ n , where n is the order, Δ − and δ − are the maximum and the minimum indegree and Δ + is the maximum outdegree of the digraph D . Some of our theorems imply well-known results on the signed 2-independence number of graphs.

Signed total (k, k)-domatic number of digraphs

Let D be a finite and simple digraph with vertex set V(D), and let f: V(D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and \({\sum_{x \in N^-(v)}f(x) \ge k}\) for each \({v \in V(G)}\), where N −(v) consists of all vertices of D from which arcs go into v, then f is a signed total k-dominating function on D. A set {f 1, f 2, . . . , f d } of signed total k-dominating functions on D with the property that \({\sum_{i=1}^df_i(x)\le k}\) for each \({x \in V(D)}\) , is called a signed total (k, k)-dominating family (of functions) on D. The maximum number of functions in a signed total (k, k)-dominating family on D is the signed total (k, k)-domatic number on D, denoted by \({d_{st}^{k}(D)}\) . In this paper we initiate the study of the signed total (k, k)-domatic number of digraphs, and we present different bounds on \({d_{st}^{k}(D)}\) . Some of our results are extensions of known properties of the signed total domatic number \({d_{st}(D)=d_{st}^{1}(D)}\) of digraphs D as well as the signed total domatic number d st (G) of graphs G, given by Henning (Ars Combin. 79:277–288, 2006).

Hardness of subgraph and supergraph problems in [formula omitted]-tournaments

Problems like the directed feedback vertex set problem have much better algorithms in tournaments when compared to general graphs. This motivates us to study a natural generalization of tournaments, named c -tournaments, and see if the structural properties of these graphs are helpful in obtaining similar algorithms. c -tournaments are simple digraphs which have directed paths of length at most c ≥ 1 between all pairs of vertices. We study the complexity of feedback vertex set and r -dominating set in c -tournaments. We also present hardness results on some graph editing problems involving c -tournaments.

On arc reversal in balanced digraphs

In this note we consider closed walks, which are cycles that are not necessarily elementary. We prove that any arc reversal in a balanced multidigraph without loops decreases the number of closed walks. This also proves that arc reversal in a simple balanced digraph decreases the number of closed walks.

FINITE TOPOLOGIES AND DIGRAPHS

In this paper we study the relation between finite topologies and digraphs. We associate a digraph to a topology by means of the “specialization” relation between points in the topology. Reciprocally, we associate a topology to each digraph, taking the sets of vertices adjacent (in the digraph) to v, for all vertex v, as a subbasis of closed sets for the topology. We use these associations to examine the relation between a simple digraph and its homologous topology. We also extend this relation to the functions preserving the structure between these classes of objects.

The signed k-domination number of directed graphs

Abstract Let k ≥ 1 be an integer, and let D = (V; A) be a finite simple digraph, for which d D− ≥ k − 1 for all v ɛ V. A function f: V → {−1; 1} is called a signed k-dominating function (SkDF) if f(N −[v]) ≥ k for each vertex v ɛ V. The weight w(f) of f is defined by $$ \sum\nolimits_{v \in V} {f(v)} $$. The signed k-domination number for a digraph D is γkS(D) = min {w(f|f) is an SkDF of D. In this paper, we initiate the study of signed k-domination in digraphs. In particular, we present some sharp lower bounds for γkS(D) in terms of the order, the maximum and minimum outdegree and indegree, and the chromatic number. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs and digraphs.

Matrix Graph Grammars with Application Conditions

In the Matrix approach to graph transformation we represent simple digraphs and rules with Boolean matrices and vectors, and the rewriting is expressed using Boolean operators only. In previous works, we developed analysis techniques enabling the study of the applicability of rule sequences, their independence, state reachability and the minimal graph able to fire a sequence. In the present paper we improve our framework in two ways. First, we make explicit (in the form of a Boolean matrix) some negative implicit information in rules. This matrix (called nihilation matrix) contains the elements that, if present, forbid the application of the rule (i.e. potential dangling edges, or newly added edges, which cannot be already present in the simple digraph). Second, we introduce a novel notion of application condition, which combines graph diagrams together with monadic second order logic. This allows for more flexibility and expressivity than previous approaches, as well as more concise conditions in certain cases. We demonstrate that these application conditions can be embedded into rules (i.e. in the left hand side and the nihilation matrix), and show that the applicability of a rule with arbitrary application conditions is equivalent to the applicability of a sequence of plain rules without application conditions. Therefore, the analysis of the former is equivalent to the analysis of the latter, showing that in our framework no additional results are needed for the study of application conditions. Moreover, all analysis techniques of [21,22] for the study of sequences can be applied to application conditions.

3-Arc-Dominated Digraphs

An oriented simple digraph $D=(V,A)$ with the minimum outdegree $d$ is called $d$-arc-dominated if for every arc $(x,y)\in A$ there is a vertex $u\in V$ with the outdegree $d$ such that both $(u,x)\in A$ and $(u,y)\in A$ hold. At the 20th British combinatorial conference, Lichiardopol posed the problem of characterizing $d$-arc-dominated digraphs. He also has posed the conjecture that a $d$-arc-dominated digraph with $d\geq2k-1$ contains $k$ vertex-disjoint directed cycles. In this paper, we give a characterization for 3-arc-dominated digraphs. Based on this characterization, we classify all 3-arc-dominated digraphs and show that the above conjecture is true when $d=3$.

ISOMORPHISMS OF TRANSFORMATION SEMIGROUPS ASSOCIATED WITH SIMPLE DIGRAPHS

For each simple digraph D, we introduce its associated semigroup S(D) and its widened digraph W(D). We show that, for any two finite simple digraphs D1 and D2, S(D1) and S(D2) are isomorphic as semigroups if and only if W(D1) and W(D2) are isomorphic as digraphs.

Contributions to Seymour’s second neighborhood conjecture

Let D be a simple digraph without loops or digons. For anyv2 V.D/ let N1.v/ be the set of all nodes at out-distance 1 from v and let N2.v/ be the set of all nodes at out-distance 2. We show that if the underlying graph is triangle-free, there must exist somev2 V.D/ such thatjN1.v/jj N2.v/j. We provide several properties a “minimal” graph which does not contain such a node must have. Moreover, we show that if one such graph exists, then there exist infinitely many.