General Digraphs Research Articles

Edge‐arc‐disjoint paths in semicomplete mixed graphs

AbstractThe so‐called weak‐2‐linkage problem asks for a given digraph and distinct vertices of whether has arc‐disjoint paths so that is an ‐path for . This problem is NP‐complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge‐disjoint and arc‐disjoint in semicomplete mixed graphs, that is, a mixed graph in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.

Coverings of general digraphs

Coverings of general digraphs

Labeling methods for partially ordered paths

The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios, and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions’ properties. Our final contribution is a big lookup table summarizing our findings and providing the reader with an easy way to choose among the most recent multi-criteria shortest path algorithms depending on their problems’ weight structure. Examples range from time-dependent shortest path bottleneck path problems to the electric vehicle shortest path problem with recharging and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and, therefore, surpass previous generalizations that were limited to acyclic graphs.

Model‐free distributed optimal control for general discrete‐time linear systems using reinforcement learning

AbstractThis article proposes a novel data‐driven framework of distributed optimal consensus for discrete‐time linear multi‐agent systems under general digraphs. A fully distributed control protocol is proposed by using linear quadratic regulator approach, which is proved to be a sufficient and necessary condition for optimal control of multi‐agent systems through dynamic programming and minimum principle. Moreover, the control protocol can be constructed by using local information with the aid of the solution of the algebraic Riccati equation (ARE). Based on the Q‐learning method, a reinforcement learning framework is presented to find the solution of the ARE in a data‐driven way, in which we only need to collect information from an arbitrary follower to learn the feedback gain matrix. Thus, the multi‐agent system can achieve distributed optimal consensus when system dynamics and global information are completely unavailable. For output feedback cases, accurate state information estimation is established such that optimal consensus control is realized. Moreover, the data‐driven optimal consensus method designed in this article is applicable to general digraph that contains a directed spanning tree. Finally, numerical simulations verify the validity of the proposed optimal control protocols and data‐driven framework.

Comparability digraphs: An analogue of comparability graphs

Comparability graphs are a popular class of graphs. We introduce as the digraph analogue of comparability graphs the class of comparability digraphs. We show that many concepts such as implication classes and the knotting graph for a comparability graph can be naturally extended to a comparability digraph. We give a characterization of comparability digraphs in terms of their knotting graphs.Semicomplete comparability digraphs contain in a sense all comparability graphs. One instrumental technique for analyzing the structure of comparability graphs is the Triangle Lemma for graphs (cf. Golumbic (1980) [7]). Using Triangle Lemma, Golumbic proved that a graph is a comparability graph if and only if no implication class of the graph is equal to its inverse. We give examples to show that a similar statement does not hold for general digraphs. We prove that it holds for semicomplete digraphs, that is, a semicomplete digraph is a comparability digraph if and only if no implication class of the digraph is equal to its inverse. The proof is done by generalizing the Triangle Lemma for graphs to semicomplete digraphs. This yields an O(n3)-time recognition algorithm for semicomplete comparability digraphs where n is the number of vertices of the input digraph. The correctness of the algorithm also implies several other characterizations for semicomplete comparability digraphs, akin to those for comparability graphs.

Oriented spanning trees and stationary distribution of digraphs

By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions. The biclique partition formula also extends the results of Knuth and Levine from line digraphs to general digraphs.

Forward and Line-Based Cycle Bases for Periodic Timetabling

The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak, and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure, and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.

Characterization of digraphs with three complementarity eigenvalues

Given a digraph D, the complementarity spectrum of the digraph is defined as the set of complementarity eigenvalues of its adjacency matrix. This complementarity spectrum has been shown to be useful in several fields, particularly in spectral graph theory. The differences between the properties of the complementarity spectrum for (undirected) graphs and for digraphs make the study of the latter of particular interest, and characterizing strongly connected digraphs with a small number of complementarity eigenvalues is a nontrivial problem. Recently, strongly connected digraphs with one and two complementarity eigenvalues have been completely characterized. In this paper, we study strongly connected digraphs with exactly three elements in the complementarity spectrum, ending with a complete characterization. This leads to a structural characterization of general digraphs having three complementarity eigenvalues.

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.

On Exponential Synchronization Rates for High-Dimensional Kuramoto Models With Identical Oscillators and Digraphs

For a high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree, although the exponential synchronization has been proved under some initial state constraints, exponential synchronization rates have not been described exactly until now. In this article, the supremum of exponential synchronization rates is precisely determined as the smallest real part of the nonzero Laplacian eigenvalues of the digraph. Our obtained result extends the existing results from the special case of strongly connected balanced digraphs to the condition of general digraphs owning directed spanning trees, which is the weakest condition for synchronization from the aspect of network structure. Moreover, our adopted method is completely different from and much more elementary than the previous differential geometry method.

Counting Homomorphic Cycles in Degenerate Graphs

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C 2k and C 2k+1 in n -vertex graphs of bounded degeneracy in time Õ( n d k ), where the fastest known algorithm for detecting directed copies of C k in general m -edge digraphs runs in time Õ( m d k ). Conversely, one can transform any O(n b k ) algorithm for computing the number of homomorphic copies of C 2k or of C 2k+1 in n -vertex graphs of bounded degeneracy, into an Õ( m b k ) time algorithm for detecting directed copies of C k in general m -edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C k -homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k -cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

Emergence of synchronization in Kuramoto model with general digraph

<p style='text-indent:20px;'>In this paper, we study the complete synchronization of the Kuramoto model with general network containing a spanning tree, when the initial phases are distributed in an open half circle. As lack of uniform coercivity in general digraph, in order to capture the dissipation structure on a general network, we apply the strongly connected graph decomposition criteria in Ha et al. [Math. Models Methods Appl. Sci., 30 (2020)] to yield a hierarchical structure, which will induce estimates similar to the hypo-coercivity estimates. This drives the phase diameter into an invariant region confined in quarter circle in finite time under a large coupling regime. Then the uniformly small boundedness of the diameter eventually leads to the emergence of exponentially fast synchronization.</p>

Cooperative Control of Multi-Time-Scale Agent Networks Under Digraphs

This paper addresses cooperative control for a class of multi-agent networks. An interesting feature of such network is that each agent possesses the characteristic of multi-time-scale. By developing the relative states or the relative outputs based consensus protocols, some well-conditioned consensus criteria in terms of linear matrix inequalities are derived successively based on the ε-strict Lyapunov function. Compared with undirected graphs or strongly connected digraphs in existing works, a more general digraph that it contains at least a directed spanning tree is considered herein. Moreover, a fully distributed consensus protocol with time-varying coupling strength is proposed to avoid using global information. Finally, the obtained theoretical results are validated by two examples including standard and nonstandard agents.

Flocking behavior of the Cucker–Smale model under a general digraph on the infinite cylinder

Flocking behavior of the Cucker–Smale model under a general digraph on the infinite cylinder

Adaptive event‐triggered bipartite consensus control for multi‐agent systems under signed digraphs

AbstractIn this paper, the bipartite consensus problem is addressed via adaptive event‐triggered control for multi‐agent systems in directed communication networks, where cooperative and competitive interactions are considered simultaneously. Under a strongly connected digraph, a novel fully distributed control protocol is first put forward, which consists of an adaptive controller and two event‐triggered mechanisms. The protocol is flexible and capable of achieving bipartite consensus without employing global information. Meanwhile, the Zeno behavior cannot occur by embedding functions in the event‐triggered mechanisms. Furthermore, a general digraph with a spanning tree is investigated, and it is shown that the proposed protocol is still effective in bipartite consensus including the leader–follower one as a special case. It is worth noting that there is no need for the protocol to continuously update the controller, nor to maintain the uninterrupted communication with neighbors. Finally, the validity of the presented results is confirmed by simulations.

Critical exponent for Cucker–Smale model under group-hierarchical multi-leadership

In this paper, we focus on the critical exponent for the Cucker–Smale model in Rd(d≥1) under group-hierarchical multi-leadership (GHML) topology. The GHML is an asymmetric topology with a group hierarchical structure and multiple leaders. The exponent β in communication weight function measures the decay rate with respect to the distance of particles. In literature, for d≥2, the critical exponent for unconditional flocking is proven to be 1/2 only for symmetric topologies or hierarchical leadership. For general digraphs, the exponent below which the unconditional flocking occurs depends on the digraph and is less than 1/2. In this paper, we prove that the critical exponent is 1/2.

On the complexity of singly connected vertex deletion

A digraph D is singly connected if for all ordered pairs of vertices u,v∈V(D), there is at most one path in D from u to v. In this paper, we study the Singly Connected Vertex Deletion (SCVD) problem: Given an n-vertex digraph D and a positive integer k, does there exist a set S⊆V(D) such that |S|≤k and D−S is singly connected? This problem may be seen as a directed counterpart of the (Undirected) Feedback Vertex Set problem, as an undirected graph is singly connected if and only if it is acyclic. SCVD is known to be NP-hard on general digraphs. We study the complexity of SCVD on various classes of digraphs such as tournaments, and various generalisations of tournaments such as digraphs of bounded independence number, in- and out-tournaments and local tournaments. We show that unlike the Feedback Vertex Set on Tournaments (FVST) problem, SCVD is polynomial-time solvable on tournaments. In addition, we show that SCVD is polynomial-time solvable on digraphs of bounded independence number, and on the class of acyclic local tournaments. We also study the parameterized complexity of SCVD, with k as the parameter, on the class of in-tournaments. And we show that on in-tournaments, SCVD admits a fixed-parameter tractable algorithm and a quadratic vertex kernel. We also show that on the class of local tournaments, which is a sub-class of in-tournaments, SCVD admits a linear vertex kernel.

Component Order Connectivity in Directed Graphs

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D=(V,A) and a pair of natural numbers k and ell , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in D-X has at most ell vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ell =1. We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, ell ,ell +k and n-ell . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^{16k}) but not in time O^*(2^{o(k)}) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^{16k}) implies the upper bound O^*(2^{16(n-ell )}) for the parameter n-ell . We complement the latter by showing that there is no algorithm of time complexity O^*(2^{o({n-ell })}) unless ETH fails. Finally, we improve (in dependency on ell ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ell +k on general digraphs from O^*(2^{O(kell log (kell ))}) to O^*(2^{O(klog (kell ))}). Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^{o(klog ell )}) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^{o(klog k)}).

On the Kernel and Related Problems in Interval Digraphs

Given a digraph G, a set \(X\subseteq V(G)\) is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set \(S\subseteq V(G)\) is said to be an independent set if no two vertices in S are adjacent in G. A kernel (resp. solution) of G is an independent and absorbing (resp. dominating) set in G. The problem of deciding if there is a kernel (or solution) in an input digraph is known to be NP-complete. Similarly, the problems of computing a minimum cardinality dominating set or absorbing set or kernel, and the problems of computing a maximum cardinality independent set or kernel, are all known to be NP-hard for general digraphs. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph G is an interval digraph if a pair of intervals \((S_u,T_u)\) can be assigned to each vertex u of G such that \((u,v)\in E(G)\) if and only if \(S_u\cap T_v\ne \emptyset \). Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs—which arise when we require that the two intervals assigned to a vertex have to intersect. We see as our main contribution the identification of the class of reflexive interval digraphs as an important class of digraphs. We show that while the problems mentioned above are NP-complete, and even hard to approximate, on interval digraphs (even on some very restricted subclasses of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate), they are all efficiently solvable, in most of the cases linear-time solvable, in the class of reflexive interval digraphs. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs. In particular, we obtain a vertex ordering characterization of reflexive interval digraphs that implies the existence of an \(O(n+m)\) time algorithm for computing a maximum cardinality independent set in a reflexive interval digraph, improving and generalizing the earlier known O(nm) time algorithm for the same problem for the interval nest digraphs. (Here m denotes the number of edges in the digraph not counting the self-loops.) We also show that reflexive interval digraphs are kernel-perfect and that a kernel in such digraphs can be computed in linear time. This generalizes and improves an earlier result that interval nest digraphs are kernel-perfect and that a kernel can be computed in such digraphs in O(nm) time. The structural characterizations that we show for point-point digraphs, apart from helping us construct the NP-completeness/APX-hardness reductions, imply that these digraphs can be recognized in linear time. We also obtain some new results for undirected graphs along the way: (a) We describe an \(O(n(n+m))\) time algorithm for computing a minimum cardinality (undirected) independent dominating set in cocomparability graphs, which slightly improves the existing \(O(n^3)\) time algorithm for the same problem by Kratsch and Stewart; and (b) We show that the Red-Blue Dominating Set problem, which is NP-complete even for planar bipartite graphs, is linear-time solvable on interval bigraphs, which is a class of bipartite (undirected) graphs closely related to interval digraphs.

Directed Steiner tree packing and directed tree connectivity

AbstractFor a digraph , and a set with and , an ‐tree is an out‐tree rooted at with . Two ‐trees and are said to be arc‐disjoint if . Two arc‐disjoint ‐trees and are said to be internally disjoint if . Let and be the maximum number of internally disjoint and arc‐disjoint ‐trees in , respectively. The generalized ‐vertex‐strong connectivity of is defined as Similarly, the generalized ‐arc‐strong connectivity of is defined as The generalized ‐vertex‐strong connectivity and generalized ‐arc‐strong connectivity are also called directed tree connectivity which extends the well‐established tree connectivity on undirected graphs to directed graphs and could be seen as a generalization of classical connectivity of digraphs. In this paper, we completely determine the complexity for both and on general digraphs, symmetric digraphs, and Eulerian digraphs. In particular, among our results, we prove and use the NP‐completeness of 2‐linkage problem restricted to Eulerian digraphs. We also give sharp bounds and equalities for the two parameters and .