General Digraphs Research Articles

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

We present emergent dynamics of continuous and discrete thermomechanical Cucker–Smale (TCS) models equipped with temperature as an extra observable on general digraph. In previous literature, the emergent behaviors of the TCS models were mainly studied on a complete graph, or symmetric connected graphs. Under this symmetric setting, the total momentum is a conserved quantity. This determines the asymptotic velocity and temperature a priori using the initial data only. Moreover, this conservation law plays a crucial role in the flocking analysis based on the elementary [Formula: see text] energy estimates. In this paper, we consider a more general connection topology which is registered by a general digraph, and the weights between particles are given to be inversely proportional to the metric distance between them. Due to this possible symmetry breaking in communication, the total momentum is not a conserved quantity, and this lack of conservation law makes the asymptotic velocity and temperature depend on the whole history of solutions. To circumvent this lack of conservation laws, we instead employ some tools from matrix theory on the scrambling matrices and some detailed analysis on the state-transition matrices. We present two sufficient frameworks for the emergence of mono-cluster flockings on a digraph for the continuous and discrete models. Our sufficient frameworks are given in terms of system parameters and initial data.

Exponential synchronization of the high-dimensional Kuramoto model with identical oscillators under digraphs

For the Kuramoto model and its variations, it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry. In this paper, for the high-dimensional Kuramoto model of identical oscillators, a matrix Riccati differential equation (MRDE) is proposed to describe the error dynamics. Based on the MRDE, the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting directed spanning trees.

Superbubbles revisited

BackgroundSuperbubbles are distinctive subgraphs in direct graphs that play an important role in assembly algorithms for high-throughput sequencing (HTS) data. Their practical importance derives from the fact they are connected to their host graph by a single entrance and a single exit vertex, thus allowing them to be handled independently. Efficient algorithms for the enumeration of superbubbles are therefore of important for the processing of HTS data. Superbubbles can be identified within the strongly connected components of the input digraph after transforming them into directed acyclic graphs. The algorithm by Sung et al. (IEEE ACM Trans Comput Biol Bioinform 12:770–777, 2015) achieves this task in mathcal {O}(m~log(m))-time. The extraction of superbubbles from the transformed components was later improved to by Brankovic et al. (Theor Comput Sci 609:374–383, 2016) resulting in an overall mathcal {O}(m+n)-time algorithm.ResultsA re-analysis of the mathematical structure of superbubbles showed that the construction of auxiliary DAGs from the strongly connected components in the work of Sung et al. missed some details that can lead to the reporting of false positive superbubbles. We propose an alternative, even simpler auxiliary graph that solved the problem and retains the linear running time for general digraph. Furthermore, we describe a simpler, space-efficient mathcal {O}(m+n)-time algorithm for detecting superbubbles in DAGs that uses only simple data structures.ImplementationWe present a reference implementation of the algorithm that accepts many commonly used formats for the input graph and provides convenient access to the improved algorithm. https://github.com/Fabianexe/Superbubble.

Disjoint paths in unions of tournaments

Given k pairs of vertices (si,ti)(1≤i≤k) of a digraph G, how can we test whether there exist vertex-disjoint directed paths from si to ti for 1≤i≤k? This is NP-complete in general digraphs, even for k=2[4], but in [3] we proved that for all fixed k, there is a polynomial-time algorithm to solve the problem if G is a tournament (or more generally, a semicomplete digraph). Here we prove that for all fixed k there is a polynomial-time algorithm to solve the problem when V(G) is partitioned into a bounded number of sets each inducing a semicomplete digraph (and we are given the partition).

Degree constrained 2-partitions of semicomplete digraphs

A 2-partition of a digraph D is a partition (V1,V2) of V(D) into two disjoint non-empty sets V1 and V2 such that V1∪V2=V(D). A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following:•For all integers k1,k2≥1 and k1+k2≥3 it is NP-complete to decide whether a given digraph D has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed choice of integers α,k1,k2≥1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed integer k≥1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition (V1,V2) such that D〈V1〉 has out-degree at least one and D〈V2〉 has in-degree at least k.•It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition (V1,V2) such that D〈Vi〉 is a strong tournament.

Consensus Conditions of Continuous-Time Multi-Agent Systems with Additive and Multiplicative Measurement Noises

This work is concerned with the consensus problem of multi-agent systems with additive and multiplicative measurement noises. By developing general stochastic stability lemmas for nonautonomous stochastic differential equations, stochastic weak and strong consensus conditions are investigated under fixed and time-varying topologies, respectively. For the case with fixed topologies and additive noises, the necessary and sufficient conditions for almost sure strong consensus are given. It is revealed that almost sure strong consensus and mean square strong consensus are equivalent under general digraphs, and almost sure weak consensus implies mean square weak consensus under undirected graphs; if multiplicative noises appear, then small noise intensities do not affect the control gain to guarantee stochastic strong consensus. For the case with time-varying topologies, sufficient consensus conditions are given under the periodically connected condition of the topology flow.

Convergence rates of discrete-time stochastic approximation consensus algorithms: Graph-related limit bounds

In this paper, we study the convergence rates of the discrete-time stochastic approximation consensus algorithms over sensor networks with communication noises under general digraphs. Basic results of stochastic analysis and algebraic graph theory are used to investigate the dynamics of the consensus error, and the mean square and sample path convergence rates of the consensus error are both given in terms of the graph and noise parameters. Especially, calculation methods to estimate the mean square limit bounds are presented under balanced digraphs, and sufficient conditions on the network topology and the step sizes are given to achieve the fast convergence rate. For the sample path limit bounds, estimation methods are also presented under undirected graphs.

Flocking of the Cucker-Smale Model on General Digraphs

In the literature, the study of the Cucker-Smale (C-S) flocking model is all restricted to connected undirected graphs or digraphs of some special types, containing only a single root. In this technical note, we establish the flocking behavior of the C-S model on general digraphs, which contain all existing interaction topologies in relation to the C-S model as special cases. In particular, the topology with multiple leaders is included. The flocking results are guaranteed under some conditions on the initial state of the flock only. The critical exponent below which unconditional convergence holds is given, depending only on the interaction topology. It is consistent with the ones obtained under some known special topologies in the literature.

Cooperative Output Regulation of Heterogeneous Nonlinear Multi-Agent Systems With Unknown Control Directions

This technical note considers global cooperative output regulation for a class of heterogeneous nonlinear multi-agent systems. By using the internal model principle and the adaptive control technique, a novel distributed adaptive control scheme is developed. Contrary to most existing results, the proposed controller is applicable to multi-agent systems with agents of non-identical relative degrees and non-identical unknown control directions under a general digraph. Moreover, any existing Nussbaum-type functions can be adopted in the proposed control scheme, and the adopted Nussbaum-type functions can be non-identical for each agent. It is shown that global cooperative output regulation of the controlled multi-agent system can be achieved under some mild assumptions. A numerical example is finally presented to illustrate the effectiveness of the proposed controller.

Polynomial kernels for deletion to classes of acyclic digraphs

We consider the problem to find a set X of vertices (or arcs) with |X|≤k in a given digraph G such that D=G−X is an acyclic digraph. In its generality, this is Directed Feedback Vertex Set (or Directed Feedback Arc Set); the existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made. In this paper, we consider the vertex deletion problem with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of the above three restrictions we can obtain a kernel with kO(1) vertices on general digraphs. We also show that the arc deletion problem with the first two restrictions (out-forest and out-tree) can be solved in polynomial time; this contrasts the status of the vertex deletion problem of these restrictions, which is NP-hard even for acyclic digraphs. The arc and vertex deletion problem with the third restriction (pumpkin), however, can be solved in polynomial time for acyclic digraphs, but becomes NP-hard for general digraphs. We believe that the idea of restricting D could yield new insights towards resolving the status of Directed Feedback Vertex Set.

Incremental single-source shortest paths in digraphs with arbitrary positive arc weights

This paper studies the incremental single-source shortest paths (SSSP) problem in general digraphs with arbitrary positive arc weights. First, we examine several properties of single-source shortest paths in general digraphs with arbitrary positive arc weights, and devise a nontrivial local search algorithm LSA to handle a single arc weight increase in such a digraph, which takes at most O(n⋅max⁡{1,nlog⁡n/m}) expected update time where n is the number of nodes and m is the number of arcs in the digraph. LSA also works on undirected graphs.Furthermore, this paper analyzes the expected update time of LSA dealing with edge weight increases or edge deletions in Erdös–Rényi (a.k.a., G(n,p)) random graphs. For weighted G(n,p) random graphs with arbitrary positive edge weights, LSA takes at most O(h(Ts)) expected update time to deal with a single edge weight increase as well as O(pn2h(Ts)) total update time, where h(Ts) is the height of input SSSP tree Ts. For G(n,p) random graphs, LSA takes O(ln⁡n) expected update time to handle a single edge deletion as well as O(pn2ln⁡n) total update time when 20ln⁡n/n≤p<2ln⁡n/n, and O(1) expected update time to handle a single edge deletion as well as O(pn2) total update time when p>2ln⁡n/n. Specifically, LSA takes the least total update time of O(nln⁡nh(Ts)) for weighted G(n,p) random graphs with p=cln⁡n/n,c>1 as well as O(n3/2(ln⁡n)1/2) for G(n,p) random graphs with p=cln⁡n/n,c>2.

On the complexity of the chip-firing reachability problem

In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in strongly polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co-NP for Eulerian digraphs.

Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.

Extremal and Degree Conditions for Path Extendability in Digraphs

In the study of cycles and paths, the meta-conjecture of Bondy that sufficient conditions for Hamiltonicity often imply pancyclicity has motivated research on the existence of cycles and paths of many lengths. Hendry further introduced the stronger concepts of cycle extendability and path extendability, which require that every cycle or path can be extended to another one with one additional vertex. These concepts have been studied extensively, but there exist few results on path extendability in digraphs, as far as we know. In this paper, we make the first attempt in this direction. We establish a number of extremal and degree conditions for path extendability in general digraphs. Moreover, we prove that every path of length at least two in a regular tournament is extendable, with some exceptions. One of our proof approaches is a new contraction operation to transform nonextendable paths into nonextendable cycles.

Chip-Firing Game and a Partial Tutte Polynomial for Eulerian Digraphs

The Chip-firing game is a discrete dynamical system played on a graph, in which chips move along edges according to a simple local rule. Properties of the underlying graph are of course useful to the understanding of the game, but since a conjecture of Biggs that was proved by Merino López, we also know that the study of the Chip-firing game can give insights on the graph. In particular, a strong relation between the partial Tutte polynomial $T_G(1,y)$ and the set of recurrent configurations of a Chip-firing game (with a distinguished sink vertex) has been established for undirected graphs. A direct consequence is that the generating function of the set of recurrent configurations is independent of the choice of the sink for the game, as it characterizes the underlying graph itself. In this paper we prove that this property also holds for Eulerian directed graphs (digraphs), a class on the way from undirected graphs to general digraphs. It turns out from this property that the generating function of the set of recurrent configurations of an Eulerian digraph is a natural and convincing candidate for generalizing the partial Tutte polynomial $T_G(1,y)$ to this class. Our work also gives some promising directions of looking for a generalization of the Tutte polynomial to general digraphs.

Proof of Berge’s path partition conjecture for [formula omitted

Let D be a digraph. A path partition of D is called k-optimal if the sum of the k-norms of its paths is minimal. The k-norm of a path P is min(|V(P)|,k). Berge’s path partition conjecture claims that for every k-optimal path partition P there are k disjoint stable sets orthogonal to P. For general digraphs the conjecture has been proven for k=1,2,λ−1,λ, where λ is the length of a longest path in the digraph. In this paper we prove the conjecture for λ−2 and λ−3.

Restricted cycle factors and arc-decompositions of digraphs

We study the complexity of finding 2-factors with various restrictions as well as edge-decompositions in (the underlying graphs of) digraphs. In particular we show that it is NP-complete to decide whether the underlying undirected graph of a digraph D has a 2-factor with cycles C1,C2,…,Ck such that at least one of the cycles Ci is a directed cycle in D (while the others may violate the orientation back in D). This solves an open problem from J. Bang-Jensen et al., Vertex-disjoint directed and undirected cycles in general digraphs, JCT B 106 (2014), 1–14. Our other main result is that it is also NP-complete to decide whether a 2-edge-colored bipartite graph has two edge-disjoint perfect matchings such that one of these is monochromatic (while the other does not have to be). We also study the complexity of a number of related problems. In particular we prove that for every even k≥2, the problem of deciding whether a bipartite digraph of girth k has a k-cycle-free cycle factor is NP-complete. Some of our reductions are based on connections to Latin squares and so-called avoidable arrays.

Strong Kernel Number in Certain Oriented Cycle Extension of Graphs

A kernel in a directed graph D(V, E) is a set S of vertices of D such that no two vertices in S are adjacent and for every vertex u in V\S there is a vertex v in S, such that (u, v) is an arc of D. The problem of existence of a kernel is itself an NP-complete for a general digraph. But in this paper we solve the strong kernel problem for certain oriented Cycle Extension of graphs namely Circular Ladder and Petersen Graphs.

Disjoint paths in tournaments

Given k pairs of vertices (si,ti)(1≤i≤k) of a digraph G, how can we test whether there exist k vertex-disjoint directed paths from si to ti for 1≤i≤k? This is NP-complete in general digraphs, even for k=2[2], but for k=2 there is a polynomial-time algorithm when G is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen [1]. Here we prove that for all fixed k there is a polynomial-time algorithm to solve the problem when G is semicomplete.

Threshold Digraphs.

A digraph whose degree sequence has a unique vertex labeled realization is called threshold. In this paper we present several characterizations of threshold digraphs and their degree sequences, and show these characterizations to be equivalent. Using this result, we obtain a new, short proof of the Fulkerson-Chen theorem on degree sequences of general digraphs.