The distribution of the domination number of class cover catch digraphs for non-uniform one-dimensional data

For two or more classes of points in R^d with d≥1, the class cover catch digraphs (CCCDs) can be constructed using the relative positions of the points from one class with respect to the points from one or all of the other classes. The CCCDs were introduced by Priebe et al. [C.E. Priebe, J.G. DeVinney, D.J. Marchette, On the distribution of the domination number of random class catch cover digraphs. Statistics and Probability Letters 55 (2001) 239-246] who investigated the case of two classes, X and Y. They calculated the exact (i.e., finite sample) distribution of the domination number of the CCCDs based on X points relative to Y points both of which were uniformly distributed on a bounded interval. We investigate the distribution of the domination number of the CCCDs based on data from non-uniform X points on an interval with end points from Y. Then we extend these calculations for multiple Y points on bounded intervals.     

Extension of one-dimensional proximity regions to higher dimensions

Proximity regions (and maps) are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which gave rise to class cover catch digraph (CCCD) and was applied to pattern classification. In this article, we note some appealing properties of the spherical proximity map in compact intervals on the real line, thereby introduce the mechanism and guidelines for defining new proximity maps in higher dimensions. For non-spherical PCDs, Delaunay tessellation (triangulation in the real plane) is used to partition the region of interest in higher dimensions. We also introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of these maps and the resulting PCDs. We provide the distribution of graph invariants, namely, domination number and relative density, of the PCDs and characterize the geometry invariance of the distribution of these graph invariants for uniform data and provide some newly defined proximity maps in higher dimensions as illustrative examples.     

Unit and proper bitolerance digraphs

In this paper we prove that the following statements about a directed graph G→ are equivalent. (1) G→ is a unit bitolerance digraph, (2) G→ is a proper bitolerance digraph, and (3) the digraph obtained by reversing all arc directions of G→ is an interval catch digraph (also known as a point-core digraph). This result combined with known algorithms for recognizing interval catch digraphs, gives the first known polynomial-time algorithm for recognizing a class of (bi)tolerance digraphs. © 1997 John Wiley & Sons, Inc.     

Cliques that are tolerance digraphs

In this paper, we study directed graph versions of tolerance graphs, in particular, the class of totally bounded bitolerance digraphs and several subclasses. When the underlying graph is complete, we prove that the classes of totally bounded bitolerance digraphs and interval catch digraphs are equal, and this implies a polynomial-time recognition algorithm for the former class. In addition, we give examples (whose underlying graphs are complete) to separate every other pair of subclasses, and one of these provides a counterexample to a conjecture of Maehara (1984).     

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.     

Locally semicomplete digraphs: A generalization of tournaments

In this paper we introduce a new class of directed graphs called locally semicomplete digraphs. These are defined to be those digraphs for which the following holds: for every vertex x the vertices dominated by x induce a semicomplete digraph and the vertices that dominate x induce a semicomplete digraph. (A digraph is semicomplete if for any two distinct vertices u and ν, there is at least one arc between them.) This class contains the class of semicomplete digraphs, but is much more general. In fact, the class of underlying graphs of the locally semi-complete digraphs is precisely the class of proper circular-arc graphs (see [13], Theorem 3). We show that many of the classic theorems for tournaments have natural analogues for locally semicomplete digraphs. For example, every locally semicomplete digraph has a directed Hamiltonian path and every strong locally semicomplete digraph has a Hamiltonian cycle. We also consider connectivity properties, domination orientability, and algorithmic aspects of locally semicomplete digraphs. Some of the results on connectivity are new, even when restricted to semicomplete digraphs.     

Biomarker discovery: classification using pooled samples

RNA-sample pooling is sometimes inevitable, but should be avoided in classification tasks like biomarker studies. Our simulation framework investigates a two-class classification study based on gene expression profiles to point out how strong the outcomes of single sample designs differ to those of pooling designs. The results show how the effects of pooling depend on pool size, discriminating pattern, number of informative features and the statistical learning method used (support vector machines with linear and radial kernel, random forest (RF), linear discriminant analysis, powered partial least squares discriminant analysis (PPLS-DA) and partial least squares discriminant analysis (PLS-DA)). As a measure for the pooling effect, we consider prediction error (PE) and the coincidence of important feature sets for classification based on PLS-DA, PPLS-DA and RF. In general, PPLS-DA and PLS-DA show constant PE with increasing pool size and low PE for patterns for which the convex hull of one class is not a cover of the other class. The coincidence of important feature sets is larger for PLS-DA and PPLS-DA as it is for RF. RF shows the best results for patterns in which the convex hull of one class is a cover of the other class, but these depend strongly on the pool size. We complete the PE results with experimental data which we pool artificially. The PE of PPLS-DA and PLS-DA are again least influenced by pooling and are low. Additionally, we show under which assumption the PLS-DA loading weights, as a measure for importance of features regarding classification, are equal for the different designs.     

Different capacities of a digraph

Sperner product is the natural generalization of co-normal product to digraphs. For every class of digraphs closed under Sperner product, the cardinality of the largest subgraph from the given class, contained as an induced subgraph in the co-normal powers of a graphG, has an exponential growth. The corresponding asymptotic exponent is the capacity ofG with respect to said class of digraphs. We derive upper and lower bounds for these capacities for various classes of digraphs, and analyze the conditions under which they are tight.     

The kth lower bases of primitive non-powerful signed digraphs

In [J. Shao, L. You, H. Shan, Bound on the bases of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 285-300], the authors extended the concept of the base from powerful sign pattern matrices to non-powerful irreducible sign pattern matrices. Recently, the kth local bases and the kth upper bases, which are generalizations of the bases, of primitive non-powerful signed digraphs were introduced. In this paper, we introduce a new parameter called the kth lower bases of primitive non-powerful signed digraphs and obtain some bounds for it. For some cases, the bounds we obtain are best possible and the extremal signed digraphs are characterized, respectively. Moreover, we show that there exist “gaps” in the kth lower bases set of primitive non-powerful signed digraphs.     

Linear arboricity of regular digraphs

A linear directed forest is a directed graph in which every component is a directed path.The linear arboricity la(D) of a digraph D is the minimum number of linear directed forests in D whose union covers all arcs of D. For every d-regular digraph D, Nakayama and P′eroche conjecture that la(D) = d + 1. In this paper, we consider the linear arboricity for complete symmetric digraphs,regular digraphs with high directed girth and random regular digraphs and we improve some wellknown results. Moreover, we propose a more precise conjecture about the linear arboricity for regular digraphs.     

Two algorithms for generating and ranking the family of solutions of a finite irreflexive relation

The family of solutions, also known as 1-bases or kernels, of a finite irreflexive relation has a variety of many interesting applications. Furthermore, the decision as towhether the associated digraph posesses a solution belongs to the class of computationally intractable problems known as NP-complete. In this paper we present (a) a tree search algorithm to generate the family of solutions of a digraph and (b) a dynamic programming algorithm to generate the family of solutions ranked in increasing order of their cardinality. Extensive computational experience with the tree search algorithm on more than 1000 randomly generated digraphs ranging from 50 to 150 vertices and from 15% to 60% densities has shown that the proposed algorithm is effective.     

On Conditional Covering Problem

The conditional covering problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facility costs required to cover all the demand points. An algorithm for CCP on paths was presented by Horne and Smith (Networks 46(4):177–185, 2005). We show that their algorithm is wrong and further present a correct O(n ³) algorithm for the same. We also propose an O(n ²) algorithm for the CCP on paths when all vertices are assigned unit costs and further extend this algorithm to interval graphs without an increase in time complexity.     

Arbitrarily traceable graphs and digraphs

The work in this paper extends and generalizes earlier work by Ore on arbitrarily traceable Euler graphs, by Harary on arbitrarily traceable digraphs, by Chartrand and White on randomly n-traversable graphs, and by Chartrand and Lick on randomly Eulerian digraphs. Arbitrarily traceable graphs of mixed type are defined and characterized in terms of a class of forbidden graphs. Arbitrarily traceable digraphs of mixed type are also defined and a simply applied characterization is given for them.     

Recognizing Knödel graphs

Knödel graphs form a class of bipartite incident-graph of circulant digraphs. This class has been extensively studied for the purpose of fast communications in networks, and it has deserved a lot of attention in this context. In this paper, we show that there exists an O(n log⁵ n)-time algorithm to recognize Knödel graphs of order 2n. The algorithm is based on a characterization of the cycles of length six in these graphs (bipartite incident-graphs of circulant digraphs always have cycles of length six). A consequence of our result is that the circulant digraphs whose chords are the power of two minus one can be recognized in O(n log⁵ n) time.     

The classification of finite groups by using iteration digraphs

A digraph is associated with a finite group by utilizing the power map f: G → G defined by f(x) = x^kfor all x ∈ G, where k is a fixed natural number. It is denoted by γG(n, k). In this paper, the generalized quaternion and 2-groups are stud- ied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2-group are determined for a 2-group to be a generalized quaternion group. Further, the classification of two generated 2-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed.     

DAG‐Width and Circumference of Digraphs

We prove that every digraph of circumference l has DAG‐width at most l. This is best possible and solves a recent conjecture from S. Kintali (ArXiv:1401.2662v1 [math.CO], January 2014).¹ As a consequence of this result we deduce that the k‐linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in J. Bang‐Jensen, F. Havet, and A. K. Maia (Theor Comput Sci 562 (2014), 283–303). We also prove that the weak k‐linkage problem (where we ask for arc‐disjoint paths) is polynomially solvable for every fixed k in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is still open. Finally, we prove that the minimum spanning strong subdigraph problem is NP‐hard on digraphs of DAG‐width at most 5.     

Role and Results of statistical methods in protein fold class prediction

Statistical methods of discrimination and classification are used for the prediction of protein structure from amino acid sequence data. This provides information for the establishment of new paradigms of carcinogenesis modeling on the basis of gene expression. Feed forward neural networks and standard statistical classification procedures are used to classify proteins into fold classes. Logistic regression, additive models, and projection pursuit regression from the family of methods based on a posterior probabilities; linear, quadratic, and a flexible discriminant analysis from the class of methods based on class conditional probabilities, and the nearest-neighbors classification rule are applied to a data set of 268 sequences. From analyzing the prediction error obtained with a test sample (n = 125) and with a cross validation procedure, we conclude that the standard linear discriminant analysis and nearest-neighbor methods are at the same time statistically feasible and potent competitors to the more flexible tools of feed forward neural networks. Further research is needed to explore the gain obtainable from statistical methods by the application to larger sets of protein sequence data and to compare the results with those from biophysical approaches.     

Chromatic Polynomials and the Symmetric Group

In this paper we give first a new combinatorial interpretation of the coefficients of chromatic polynomials of graphs in terms of subsets of permutations. Motivated by this new interpretation, we introduce next a combinatorially defined polynomial associated to a directed graph, and prove that it is related to chromatic polynomials. These polynomials are a specialization of cover polynomials of digraphs.I am grateful to the Swiss National Science Foundation for its partial financial supportFinal version received: June 25, 2003     

A Spectral Study of the Manhattan Networks

The multidimensional Manhattan networks are a family of digraphs with many appealing properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we fully determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity.     

On the structure of strong 3-quasi-transitive digraphs

In this paper, D=(V(D),A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V(D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u→v→w→z in D, then u and z are adjacent or u=z. In Bang-Jensen (2004) [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.