Queue layouts of iterated line directed graphs

In this paper, we study queue layouts of iterated line directed graphs. A k-queue layout of a directed graph consists of a linear ordering of the vertices and an assignment of each arc to exactly one of the k queues so that any two arcs assigned to the same queue do not nest. The queuenumber of a directed graph is the minimum number of queues required for a queue layout of the directed graph.We present upper and lower bounds on the queuenumber of an iterated line directed graph L^k(G) of a directed graph G. Our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the results on the queuenumber of L^k(G), it is shown that for any fixed directed graph G, L^k(G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L^k(G). These results are also applied to specific families of iterated line directed graphs such as de Bruijn, Kautz, butterfly, and wrapped butterfly directed graphs. In particular, the queuenumber of k-ary butterfly directed graphs is determined if k is odd.     

Directed t-Packings and Directed t-Steiner Systems

The aim of this paper is an investigation of directed t-packings and in particular of directed t-Steiner systems. A new upper bound on the number of points k for directed t-Steiner systems T(t,k,k) is obtained. We disprove a conjecture of Levenshtein on T(t,k,k) for t 3 by showing that a T(4,6,6) exists. Furthermore, it is proved that the symmetric group S ₆ can be partitioned into 30 disjoint T(4,6,6)s. Extensive computer search shows that the tight upper bound on K for t =4,5 is 6 and for t=6 is 7. The non-existence of further small directed t-Steiner systems is established, and large directed t-packings for t,4,5,6 are constructed.     

Feasibility of motion planning on acyclic and strongly connected directed graphs

Motion planning is a fundamental problem of robotics with applications in many areas of computer science and beyond. Its restriction to graphs has been investigated in the literature, for it allows one to concentrate on the combinatorial problem abstracting from geometric considerations. In this paper, we consider motion planning over directed graphs, which are of interest for asymmetric communication networks. Directed graphs generalize undirected graphs, while introducing a new source of complexity to the motion planning problem: moves are not reversible. We first consider the class of acyclic directed graphs and show that the feasibility can be solved in time linear in the product of the number of vertices and the number of arcs. We then turn to strongly connected directed graphs. We first prove a structural theorem for decomposing strongly connected directed graphs into strongly biconnected components. Based on the structural decomposition, we show that the feasibility of motion planning on strongly connected directed graphs can be decided in linear time.     

The Spectrum of （gv,g, 3, λ）-dDF in Zgv

In this paper, several recursive constructions for directed difference family and perfect directed difference family are presented by means of difference matrix and incomplete difference matrix. Finally the necessary and sufficient conditions for the existence of a （gv, g, 3, λ）-directed difference family in Zgv are established. As a consequence, the necessary and sufficient conditions for the existence of a cyclic directed group divisible design with block size three and type gv are obtained.     

Almost Given Length Cycles in Digraphs

A digraph is called k-cyclic if it cannot be made acyclic by removing less than k arcs. It is proved that for every ε > 0 there are constants K and δ so that for every d ∈ (0, δn), every ε n²-cyclic digraph with n vertices contains a directed cycle whose length is between d and d + K. A more general result of the same form is obtained for blow-ups of directed cycles.     

Signed domination numbers of directed graphs

The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.     

Directoid groups

We continue the study of directoid groups, directed abelian groups equipped with an extra binary operation which assigns an upper bound to each ordered pair subject to some natural restrictions. The class of all such structures can to some extent be viewed as an equationally defined substitute for the class of (2-torsion-free) directed abelian groups. We explore the relationship between the two associated categories, and some aspects of ideals of directoid groups.     

The Category of Directed Systems in a Category

We present two related categorical constructions. Given a category C, we construct a category C^[d], the category of directed systems in C. C embeds into C^[d], and if C has enough colimits, then C is monadic over C^[d]. Also, if E,M is a factorization structure for C, then C^[d] has a related factorization structure E_d M_d such that if E consists entirely of monic arrows, then so does E_d and the E_d-quotient poset of an object A is naturally the poset of directed downsets of the E-quotient poset of A. Similarly, if M consists entirely of monicarrows, then so does M_d and the M_d-subobject poset of an object A is naturally the poset of directed downsets of the M-subobject poset. C^[d] has completeness and cocompleteness properties at least as good as those of C, and it is abelian if C is. Dualization gives the other construction: a category C^[i], the category of inverse systems in C, into which C also embeds and which satisfies similar properties, except that directed downsets in the E-quotient and M-subobject posets are replaced by directed upsets.     

Induced good gradings of structural matrix rings

Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group G, a G-grading set is a set of arrows with the property that any assignment of these arrows to elements of G uniquely determines an induced good grading. One of our main results is that a G-grading set exists for any transitive directed graph if G is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which do not have G-grading sets for any cyclic group G of even order greater than 2. Finally, we count the number of nonequivalent elementary gradings by a finite group of a full matrix ring over an arbitrary field.     

Maximum directed cuts in digraphs with degree restriction

For integers m, k≥1, we investigate the maximum size of a directed cut in directed graphs in which there are m edges and each vertex has either indegree at most k or outdegree at most k. © 2009 Wiley Periodicals, Inc. J Graph Theory     

Directed submodularity,ditroids and directed submodular flows

Set relations and operations such as inclusion, union and intersection are generalized to directed subsets whose elements are distinguished between forward and backward elements. The concepts of submodular functions, matroids and polymatroidal network flows are extended to the concepts of directed submodular functions, ditroids and directed submodular flows on directed subsets. Two unrelated matroids (submodular functions) can be embedded in one ditroid (directed submodular function). Total dual integrality is preserved in these generalizations and proved for very general set-function class-directed odd submodular functions.This work was partially supported by Chinese National Natural Science Fund.     

The acircuitic directed star arboricity of subcubic graphs is at most four

A directed star forest is a forest all of whose components are stars with arcs emanating from the center to the leaves. The acircuitic directed star arboricity of an oriented graph G (that is a digraph with no opposite arcs) is the minimum number of arc-disjoint directed star forests whose union covers all arcs of G and such that the union of any two such forests is acircuitic. We show that every subcubic graph has acircuitic directed star arboricity at most four.     

(向量)R_n和(向量)G_n的有向优美性

优美图可用在图论中的某些H-分解问题中,很多人研究无向图的优美标号.研究有向优美标号,通过对阶数奇偶性的讨论,给出了n(≥2)阶有向路(向量)P_n和n(≥3)阶有向(向量)C_n圈是有向优美的充分条件.     

战争地区邮递员问题

本文首次提出了赋权有向图上中国邮递员问题的一个推广－战争地区邮递员问题，并对解的存在性给出了若干充分条件和必要条件，得到了求解该问题的一个多项式算法。     

Coloring all directed paths in a symmetric tree,with an application to optical networks*

Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colors needed to color the set of all directed paths in T, so that two paths of the same color never use the same directed arc of T, is equal to the maximum number of different paths that contain the same arc of T. The proof implies a polynomial time algorithm for actually coloring the paths with the minimum number of colors. When only a subset of the directed paths is to be colored, the problem is known to be NP‐complete; we describe certain instances of the problem which can be efficiently solved. These results are applied to WDM (wavelength‐division multiplexing) routing in all‐optical networks. In particular, we solve the all‐to‐all gossiping problem in optical networks. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 183–196, 2001     

具有最小弧数的唯一泛圈有向图的计数

唯一泛圈有向图D是一个定向图,对每一个n,3≤n≤υ,D中有且只有一个长为n的有向圈.用g(υ)表示具有υ个顶点的唯一泛圈有向图最小可能的弧数,用N(υ)表示具有υ个顶点、g(υ)条弧且互不同构的唯一泛圈有向图的个数.确定了当υ=3,4,5,6,7,8时的N(υ).     

有向圈的行列式算法及HAMILTON图条件

本文引入有向路乘法、弧行列式等概念 ,讨论了弧行列式的性质 ,阐述了二种计算有向圈的行列式方法及有向图 D为 Hamilton图的充要条件 ,最后给出了计算实例     

Characterizing directed path graphs by forbidden asteroids

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it does not contain an asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a special connection. Two non‐adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex‐disjoint chordless paths satisfying certain conditions. A special asteroidal triple is an asteroidal triple such that each pair is linked by a special connection. We prove that a chordal graph is a directed path graph if and only if it does not contain a special asteroidal triple. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:103‐112, 2011     

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets

This paper is a continuation of the author's first paper (Set-Valued Anal. 9 (2001), pp. 217–245), where the normed and partially ordered vector space of directed sets is constructed and the cone of all nonempty convex compact sets in R ⁿ is embedded. A visualization of directed sets and of differences of convex compact sets is presented and its geometrical components and properties are studied. The three components of the visualization are compared with other known differences of convex compact sets.     

关于定向极小集

本文对定向极小集作了进一步的研究，得到一系列重要性质，文章最后给出连续格为完全分配格的一个充分条件．