A Characterization of Mixed Branching Greedoids

Branching greedoids have been defined and characterized for both directed and undirected rooted graphs. Such greedoids can be extended to rooted mixed graphs – graphs with both directed and undirected edges. These greedoids are characterized by a list of forbidden minors. If Ω is a rooted mixed graph, its mixed branching greedoid has the edges of Ω as its ground set and the collection of arborescences as its feasible sets. The set of mixed branching greedoids is exactly the set of local forest greedoids without

Polymatroid greedoids

This paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far, are demonstrated.     

In and out forests on combinatorial landscapes

Fitness landscape theory is a mathematical framework for numerical analysis of search algorithms on combinatorial optimization problems. We study a representation of fitness landscape as a weighted directed graph. We consider out forest and in forest structures in this graph and establish important relationships among the forest structures of a directed graph, the spectral properties of the Laplacian matrices, and the numbers of local optima of the landscape. These relationships provide a new approach for computing the numbers of local optima for various problem instances and neighborhood structures.     

冠状系统的R-旋转图与(R)-旋转图

冠状系统Hc的R(R)-旋转变换是指对Hc的一个完美匹配M,同时将Hc中所有正常(非正常)M-交错的六边形变换为非正常(正常)M-交错的六边形,从而得到Hc的另一个完美匹配的变换.通过这两种旋转变换可分别建立Hc完美匹配集上的层次结构,分别称为R-旋转图和R-旋转图,记为R(Hc)和R(Hc).已经证明知道R(Hc)是有向森林,其每个分支都为有向根树.首先讨论了冠状系统的Z-变换有向图与其R-旋转图之间的关系,指出按连通分支对这两种图的顶点集进行划分,其结果一样.在此基础上,证明了R(Hc)的任一分支T(有向根树)都对应R(Hc)的一个分支T,且两者的顶点集相同,进而证明了T与(T)具有相同的高度和宽度.     

Relations between the Local Chromatic Number and Its Directed Version

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm.     

Explicit Construction and Uniqueness for Universal Operator Algebras of Directed Graphs

Given a directed graph, there exist a universal operator algebraand universal C*-algebra associated to the directed graph. Inthis paper we give intrinsic constructions for these objects.We also provide an explicit construction for the maximal C*-algebraof an operator algebra. We discuss uniqueness of the universalalgebras for finite graphs, showing that for finite graphs thegraph is an isomorphism invariant for the universal operatoralgebra of a directed graph. We show that the underlying undirectedgraph is a Banach algebra isomorphism invariant for the universalC*-algebra of a directed graph.     

On Brylawski’s Generalized Duality

We introduce a notion of duality??due to Brylawski??that generalizes matroid duality to arbitrary rank functions. This allows us to define a generalization of the matroid Tutte polynomial. This polynomial satisfies a deletion-contraction recursion, where deletion and contraction are defined in this more general setting. We explore this notion of duality for greedoids, antimatroids and demi-matroids, proving that matroids correspond precisely to objects that are simultaneously greedoids and ??dual?? greedoids.     

Homomorphisms and amalgamation

We prove that for each finite core graph G, the class of all graphs admitting a homomorphism into G is a pseudo-amalgamation class, in the sense of Fraı̈ssé. This fact gives rise to a countably infinite universal pseudo-homogeneous graph which shares some of the properties of the infinite random graph. Our methods apply simultaneously to G-colourings in several classes of relational structures, such as the classes of directed graphs or hypergraphs.     

Order Dimension of Orthomodular Amalgamations Over Trees

We consider the amalgamation of bounded involution posets over a strictly directed graph as applied to orthomodular lattices, orthomodular posets or orthoalgebras. In the finite setting, we show that the order dimension of the amalgamation does not exceed that of the amalgamated structures by more than one. We also present conditions under which equality obtains.      

A mixed directed-undirected data structure for a parallel implementation of a domain decomposition algorithm

The choice of data structures influences the parallelization, efficiency and the manageability of a mesh refinement program. We introduce a mixed directed-undirected graph that combines both communication and scheduling needs. An inverted index is maintained for the directed graph to improve code performance and readability.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.     

Locally-finite connected-homogeneous digraphs

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.     

The Myerson value for directed graph games

A directed graph game consists of a cooperative game with transferable utility and a digraph which describes limited cooperation and the dominance relation among the players. Under the assumption that only coalitions of strongly connected players are able to fully cooperate, we introduce the digraph-restricted game in which a non-strongly connected coalition can only realize the sum of the worths of its strong components. The Myerson value for directed graph games is defined as the Shapley value of the digraph-restricted game. We establish axiomatic characterizations of the Myerson value for directed graph games by strong component efficiency and either fairness or bi-fairness.     

Minimal comparability completions of arbitrary graphs

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes Π for which Π completion of arbitrary graphs can be achieved through such a vertex incremental approach.     

A unified framework for harmonic analysis of functions on directed graphs and changing data

We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner.     

The spectral radii of a graph and its line graph

The spectral radius of a (directed) graph is the largest eigenvalue of adjacency matrix of the (directed) graph. We give the relation on the characteristic polynomials of a directed graph and its line graph, and obtain sharp bounds on the spectral radius of directed graphs. We also give the relation on the spectral radii of a graph and its line graph. As a consequence, the spectral radius of a connected graph does not exceed that of its line graph except that the graph is a path.     

Bounding the feedback vertex number of digraphs in terms of vertex degrees

The Turán bound (Turán (1941) [17]) is a famous result in graph theory, which relates the independence number of an undirected graph to its edge density. Also the Caro-Wei inequality (Caro (1979) [4] and Wei (1981) [18]), which gives a more refined bound in terms of the vertex degree sequence of a graph, might be regarded today as a classical result. We show how these statements can be generalized to directed graphs, thus yielding a bound on directed feedback vertex number in terms of vertex out-degrees and in terms of average out-degree, respectively.     

Minimum Spanning Table and Optimal Expansion of Competence Set

Each directed graph with asymmetric costs defined over its arcs can be represented by a matrix or table, called an expansion table. We explore first the basic properties of cycles and spanning tables of expansion tables, which correspond to the cycles and spanning trees of the directed graph. Then, we derive an algorithm to find a minimum spanning table which corresponds to a minimum spanning tree in the directed graph. Finally, we discuss how to use the algorithm to find the optimal competence set expansion and also discuss related problems.     

Branchings in rooted graphs and the diameter of greedoids

LetG be a 2-connected rooted graph of rankr andA, B two (rooted) spanning trees ofG We show that the maximum number of exchanges of leaves that can be required to transformA intoB isr ²–r+1 (r>0). This answers a question by L. Lovász.There is a natural reformulation of this problem in the theory ofgreedoids, which asks for the maximum diameter of the basis graph of a 2-connected branching greedcid of rankr.Greedoids are finite accessible set systems satisfying the matroid exchange axiom. Their theory provides both language and conceptual framework for the proof. However, it is shown that for general 2-connected greedoids (not necessarily constructed from branchings in rooted graphs) the maximum diameter is 2^r–1.     

Asteroids in rooted and directed path graphs

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 contains no 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 strong path. Two non-adjacent vertices are linked by a strong path if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain conditions. A strong asteroidal triple is an asteroidal triple such that each pair is linked by a strong path. We prove that a chordal graph is a directed path graph if and only if it contains no strong asteroidal triple. We also introduce a related notion of asteroidal quadruple, and conjecture a characterization of rooted path graphs which are the intersection graphs of directed paths in a rooted tree.