Algebraic theory for the clique operator

In this text we attempt to unify many results about the K operator based on a new theory involving graphs, families and operators. We are able to build an "operator algebra" that helps to unify and automate arguments. In addition, we relate well-known properties, such as the Helly property, to the families and the operators. As a result, we deduce many classic results in clique graph theory from the basic fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Bandelt and Prisner's partial characterization of clique-fixed classes [2]. Furthermore, we show the power of our approach proving general results that lead to polynomial recognition of certain graph classes.Facultad de Ciencias Exacta

Algebraic theory for the clique operator

In this text we attempt to unify many results about the K operator based on a new theory involving graphs, families and operators. We are able to build an "operator algebra" that helps to unify and automate arguments. In addition, we relate well-known properties, such as the Helly property, to the families and the operators. As a result, we deduce many classic results in clique graph theory from the basic fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Bandelt and Prisner's partial characterization of clique-fixed classes [2]. Furthermore, we show the power of our approach proving general results that lead to polynomial recognition of certain graph classes.Facultad de Ciencias Exacta

Least Squares Ranking on Graphs

Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data. Since an exact match will usually be impossible, one settles for matching in a least squares sense. This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook. If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al. With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch. These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams). Not all of these connections are explored in this paper, but many are. The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra. One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic. Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development.Comment: Added missing references, comparison of linear solvers overhauled, conclusion section added, some new figures adde

HH-product and HH-threshold graphs

This paper is the continuation of the research of the author and his colleagues of the {\it canonical} decomposition of graphs. The idea of the canonical decomposition is to define the binary operation on the set of graphs and to represent the graph under study as a product of prime elements with respect to this operation. We consider the graph together with the arbitrary partition of its vertex set into $n$ subsets ($n$-partitioned graph). On the set of $n$-partitioned graphs distinguished up to isomorphism we consider the binary algebraic operation $\circ_H$ ($H$-product of graphs), determined by the digraph $H$. It is proved, that every operation $\circ_H$ defines the unique factorization as a product of prime factors. We define $H$-threshold graphs as graphs, which could be represented as the product $\circ_{H}$ of one-vertex factors, and the threshold-width of the graph $G$ as the minimum size of $H$ such, that $G$ is $H$-threshold. $H$-threshold graphs generalize the classes of threshold graphs and difference graphs and extend their properties. We show, that the threshold-width is defined for all graphs, and give the characterization of graphs with fixed threshold-width. We study in detail the graphs with threshold-widths 1 and 2

Byzantine Approximate Agreement on Graphs

Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors\u27 input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures