Decompositions Of Digraphs Research Articles

Strong arc decompositions of split digraphs

AbstractA strong arc decomposition of a digraph is a partition of its arc set into two sets such that the digraph is strong for . Bang‐Jensen and Yeo conjectured that there is some such that every ‐arc‐strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2‐arc‐strong semicomplete digraph has a strong arc decomposition. Bang‐Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2‐arc‐strong locally semicomplete digraph which is not the square of an even cycle has a strong arc decomposition. This implies that every 3‐arc‐strong locally semicomplete digraph has a strong arc decomposition. A split digraph is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partitioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph by adding a new set of vertices and some arcs between and . In this paper, we prove that every 3‐arc‐strong split digraph has a strong arc decomposition which can be found in polynomial time and we provide infinite classes of 2‐strong split digraphs with no strong arc decomposition. We also pose a number of open problems on split digraphs.

Decompositions of the λ-Fold Complete Mixed Graph into Mixed 6-Stars

Graph and digraph decompositions are a fundamental part of design theory. Probably the best known decompositions are related to decomposing the complete graph into 3-cycles (which correspond to Steiner triple systems), and decomposing the complete digraph into orientations of a 3-cycle (the two possible orientations of a 3-cycle correspond to directed triple systems and Mendelsohn triple systems). Decompositions of the λ-fold complete graph and the λ-fold complete digraph have been explored, giving generalizations of decompositions of complete simple graphs and digraphs. Decompositions of the complete mixed graph (which contains an edge and two distinct arcs between every two vertices) have also been explored in recent years. Since the complete mixed graph has twice as many arcs as edges, an isomorphic decomposition of a complete mixed graph into copies of a sub-mixed graph must involve a sub-mixed graph with twice as many arcs as edges. A partial orientation of a 6-star with two edges and four arcs is an example of such a mixed graph; there are five such mixed stars. In this paper, we give necessary and sufficient conditions for a decomposition of the λ-fold complete mixed graph into each of these five mixed stars for all λ>1.

Вычислительная сложность оптимального блокирования вершин в орграфе

In this paper, we solve the problem of determining the minimum set of edges, whose removal from the digraph breaks all paths, that pass through the selected set of vertices. This problem is reduced to the problem of the minimum section and maximum flow in a bipolar junction. Methods of digraph decomposition that reduce its computational complexity are proposed.

Decompositions of Edge-Colored Digraphs: A New Technique in the Construction of Constant-Weight Codes and Related Families

We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes and multiply constant-weight codes. This was achieved via an interesting application of the theory of decomposition of edge-colored digraphs.

K-distinct in- and out-branchings in digraphs

An out-branching and an in-branching of a digraph D are called k-distinct if each of them has k arcs absent in the other. Bang-Jensen, Saurabh and Simonsen (2016) proved that the problem of deciding whether a strongly connected digraph D has k-distinct out-branching and in-branching is fixed-parameter tractable (FPT) when parameterized by k. They asked whether the problem remains FPT when extended to arbitrary digraphs. Bang-Jensen and Yeo (2008) asked whether the same problem is FPT when the out-branching and in-branching have the same root. By linking the two problems with the problem of whether a digraph has an out-branching with at least k leaves (a leaf is a vertex of out-degree zero), we first solve the problem of Bang-Jensen and Yeo (2008). We then develop a new digraph decomposition and using it prove that the problem of Bang-Jensen et al. (2016) is FPT for all digraphs.

Path decompositions of digraphs and their applications to Weyl algebra

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the n-th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem and helps to study skew-symmetric polynomials on certain subspaces of Weyl algebra. For instance, path decompositions can be used to study minimal polynomial identities on Weyl algebra, similarly as Eulerian tours applicable for Amitsur–Levitzki theorem. We introduce the G-Stirling functions which enumerate decompositions by sources (and sinks) of paths.

On the monotonicity of process number

Graph searching games involve a team of searchers that aims at capturing a fugitive in a graph. These games have been widely studied for their relationships with the tree- and the path-decomposition of graphs. In order to define decompositions for directed graphs, similar games have been proposed in directed graphs. In this paper, we consider a game that has been defined and studied in the context of routing reconfiguration problems in WDM networks. Namely, in the processing game, the fugitive is invisible, arbitrarily fast, it moves in the opposite direction of the arcs of a digraph, but only as long as it can access to a strongly connected component free of searchers. We prove that the processing game is monotone which leads to its equivalence with a new digraph decomposition.

On the Monotonicity of Process Number

Graph searching games involve a team of searchers that aims at capturing a fugitive in a graph. These games have been widely studied for their relationships with tree-and path-decomposition of graphs. In order to define decompositions for directed graphs, similar games have been proposed in directed graphs. In this paper, we consider such a game that has been defined and studied in the context of routing reconfiguration problems in WDM networks. Namely, in the processing game, the fugitive is invisible, arbitrary fast, it moves in the opposite direction of the arcs of a digraph, but only as long as it has access to a strongly connected component free of searchers. We prove that the processing game is monotone which leads to its equivalence with a new digraph decomposition.

Digraph decompositions and monotonicity in digraph searching

Graph decompositions such as tree-decompositions and associated width measures have been the focus of much attention in structural and algorithmic graph theory. In particular, it has been found that many otherwise intractable problems become tractable on graph classes of bounded tree-width. More recently, proposals have been made to define a similar notion to tree-width for directed graphs. Several proposals have appeared so far, supported by algorithmic applications. In this paper we explore the limits of algorithmic applicability of digraph decompositions and show that various natural candidates for problems, which potentially could benefit from digraphs having small “directed width”, remain NP-complete even on almost acyclic graphs. Closely related to graph and digraph decompositions are graph searching games. An important property of graph searching games is monotonicity and a large number of papers addresses the question whether particular variants of these games are monotone. However, so far for two natural types of graph searching games–underlying DAG- and Kelly-decompositions–the question whether they are monotone was still open. We settle this issue by showing that both variants, the visible and the inert invisible graph searching games on directed graphs, are non-monotone.

On the algorithmic effectiveness of digraph decompositions and complexity measures

We place our focus on the gap between treewidth’s success in producing fixed-parameter polynomial algorithms for hard graph problems, and specifically Hamiltonian Circuit and Max Cut, and the failure of its directed variants (directed treewidth (Johnson et al., 2001 [13]), DAG-width (Obdrzálek, 2006 [14]) and Kelly-width (Hunter and Kreutzer, 2007 [15]) to replicate it in the realm of digraphs. We answer the question of why this gap exists by giving two hardness results: we show that Directed Hamiltonian Circuit is W [ 2 ] -hard when the parameter is the width of the input graph, for any of these widths, and that Max Di Cut remains NP-hard even when restricted to DAGs, which have the minimum possible width under all these definitions. Along the way, we extend our reduction for Directed Hamiltonian Circuit to show that the related Minimum Leaf Outbranching problem is also W [ 2 ] -hard when naturally parameterized by the number of leaves of the solution, even if the input graph has constant width. All our results also apply to directed pathwidth and cycle rank.

On relation between spectra of graphs and their digraph decompositions

A graph, consisting of undirected edges, can be represented as a sum of two digraphs, consisting of oppositely oriented directed edges. Gutman and Plath in [J. Serb. Chem. Soc. 66 (2001), 237-241] showed that for annulenes, the eigenvalue spectrum of the graph is equal to the sum of the eigenvalue spectra of respective two digraphs. Here we exhibit a number of other graphs with this property.

Digraph measures: Kelly decompositions, games, and orderings

We consider various well-known, equivalent complexity measures for graphs such as elimination orderings, k -trees and cops and robber games and study their natural translations to digraphs. We show that on digraphs the translations of these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digraphs and an associated width, Kelly-width, which is equivalent to the aforementioned measure. We demonstrate its usefulness by exhibiting potential applications including polynomial-time algorithms for NP-complete problems on graphs of bounded Kelly-width, and complexity analysis of asymmetric matrix factorization. Finally, we compare the new width to other known decompositions of digraphs.

Bounded degree acyclic decompositions of digraphs

An acyclic decomposition of a digraph is a partition of the edges into acyclic subgraphs. Trivially every digraph has an acyclic decomposition into two subgraphs. It is proved that for every integer s⩾2 every digraph has an acyclic decomposition into s subgraphs such that in each subgraph the outdegree of each vertex v is at most deg(v) s−1 . For all digraphs this degree bound is optimal.

Orthogonal Decompositions of Complete Digraphs

A family ? of isomorphic copies of a given digraph \(\) is said to be an orthogonal decomposition of the complete digraph \(\) by \(\), if every arc of \(\) belongs to exactly one member of ? and the union of any two different elements from ? contains precisely one pair of reverse arcs.

TSP tour domination and Hamilton cycle decompositions of regular digraphs

In this paper, we solve a problem by Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502–510) from the context of domination analysis, where the performance of a heuristic algorithm is rated by the number of solutions that are not better than the solution found by the algorithm, rather than by the relative performance compared to the optimal value. In particular, we show that for the asymmetric traveling salesman problem, there is a deterministic polynomial time algorithm that finds a tour that is at least as good as the median of all tour values. Our algorithm uses an unpublished theorem by Häggkvist on the Hamilton decomposition of regular digraphs.

Hamiltonian cycles and decompositions of cayley digraphs of finite abelian groups

In this paper, Hamiltonian cycles and decompositions of Cayley digraphs are investigated. Sufficient conditions are given for these two problems respectively. Furthermore, the conditions are also necessary for 2-regular Cayley digraphs. In addition, some known results about the Cartesian products of two directed cycles are also deduced.

Digraph Decompositions and Eulerian Systems

The theory of digraph decompositions introduced by W. Cunningham and the theory of isotropic systems introduced by the author are unified. A basic combinatorial tool is the operation of local complementation at a vertex of a digraph, a generalization of the similar operation already known for simple graphs. This allows us to unify in a single class the semibrittle digraphs characterized by W. Cunningham and to devise a more efficient algorithm for searching for a split of a digraph.

Applying a proof of tverberg to complete bipartite decompositions of digraphs and multigraphs

AbstractGraham and Pollak [2] proved that n – 1 is the minimum number of edge‐disjoint complete bipartite subgraphs into which the edges of Kn decompose. Tverberg [6], using a linear algebraic technique, was the first to give a simple proof of this result. We apply Tverberg's technique to obtain results for two related decomposition problems, in which we wish to partition the arcs/edges of complete digraphs/multigraphs into a minimum number of arc/edge‐disjoint complete bipartite subgraphs of appropriate natures. We obtain exact results for the digraph problem, which was posed by Lundgren and Maybee [4]. We also obtain exact results for the decomposition of complete multigraphs with exactly two edges connecting every pair of distinct vertices.

A note about a theorem by maamoun on decompositions of digraphs into elementary directed paths or cycles

Maamoun ( J. Combin. Theory Ser. B 38 (1985), 97–101) has found an upper bound on the minimum number of elementary directed paths or elementary directed cycles which can partition the arcs of a digraph G. We slightly improve his theorem by specifying the number of elementary directed paths needed in such a partition.

Decompositions of digraphs into paths and cycles

It is shown that in a digraph G, there is an elementary directed path or an elementary directed cycle meeting all inclusion-maximal demi-cocycles of G. This theorem is used to obtain an upper bound on the cardinality of a minimal partition of the arc set of G into directed paths and cycles.