Direct Product Primality Testing of Graphs is GI-hard

We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem

Recognizing Cartesian graph bundles

AbstractGraph bundles generalize the notion of covering graphs and graph products. In this paper we extend some of the methods for recognizing Cartesian product graphs to graph bundles. Two main notions are used. The first one is the well-known equivalence relation δ★ defined on the edge-set of a graph. The second one is the concept of k-convex subgraphs. A subgraph H is k-convex in G, if for any two vertices x and y of distance d, d ⩽ k, each shortest path from x to y in G is contained entirely in H. The main result is an algorithm that finds a representation as a nontrivial Cartesian graph bundle for all graphs that are Cartesian graph bundles over a triangle-free simple base. The problem of recognizing graph bundles over a base containing triangles remains open

Factoring cardinal product graphs in polynomial time

AbstractIn this paper a polynomial algorithm for the prime factorization of finite, connected nonbipartite graphs with respect to the cardinal product is presented. This algorithm also decomposes finite, connected graphs into their prime factors with respect to the strong product and provides the basis for a new proof of the uniqueness of the prime factorization of finite, connected nonbipartite graphs with respect to the cardinal product. Furthermore, some of the consequences of these results and several open problems are discussed

The Complement of the Djokovic-Winkler Relation

The Djokovi\'{c}-Winkler relation $\Theta$ is a binary relation defined on the edge set of a given graph that is based on the distances of certain vertices and which plays a prominent role in graph theory. In this paper, we explore the relatively uncharted ``reflexive complement'' $\overline\Theta$ of $\Theta$, where $(e,f)\in \overline\Theta$ if and only if $e=f$ or $(e,f)\notin \Theta$ for edges $e$ and $f$. We establish the relationship between $\overline\Theta$ and the set $\Delta_{ef}$, comprising the distances between the vertices of $e$ and $f$ and shed some light on the intricacies of its transitive closure $\overline\Theta^*$. Notably, we demonstrate that $\overline\Theta^*$ exhibits multiple equivalence classes only within a restricted subclass of complete multipartite graphs. In addition, we characterize non-trivial relations $R$ that coincide with $\overline\Theta$ as those where the graph representation is disconnected, with each connected component being the (join of) Cartesian product of complete graphs. The latter results imply, somewhat surprisingly, that knowledge about the distances between vertices is not required to determine $\overline\Theta^*$. Moreover, $\overline\Theta^*$ has either exactly one or three equivalence classes

Robust Factorizations and Colorings of Tensor Graphs

Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around SDP-based rounding. However, it is likely that important combinatorial or algebraic insights are needed in order to break the $n^{o(1)}$ threshold. One way to develop new understanding in graph coloring is to study special subclasses of graphs. For instance, Blum studied the 3-coloring of random graphs, and Arora and Ge studied the 3-coloring of graphs with low threshold-rank. In this work, we study graphs which arise from a tensor product, which appear to be novel instances of the 3-coloring problem. We consider graphs of the form $H = (V,E)$ with $V =V( K_3 \times G)$ and $E = E(K_3 \times G) \setminus E'$, where $E' \subseteq E(K_3 \times G)$ is any edge set such that no vertex has more than an $\epsilon$ fraction of its edges in $E'$. We show that one can construct $\widetilde{H} = K_3 \times \widetilde{G}$ with $V(\widetilde{H}) = V(H)$ that is close to $H$. For arbitrary $G$, $\widetilde{H}$ satisfies $|E(H) \Delta E(\widetilde{H})| \leq O(\epsilon|E(H)|)$. Additionally when $G$ is a mild expander, we provide a 3-coloring for $H$ in polynomial time. These results partially generalize an exact tensor factorization algorithm of Imrich. On the other hand, without any assumptions on $G$, we show that it is NP-hard to 3-color $H$.Comment: 34 pages, 3 figure

Using factor score estimates in latent variable analysis

This dissertation consists of 2 separate papers. Both papers include topics related to using factor score estimates in latent variable analysis.;Here is an abstract for the paper entitled: Nonlinear latent covariate analysis using factor score estimate. Latent variables have an important role in assessing the effectiveness of comparative treatment outcomes in social and behavioral studies. In such studies, the latent intervention treatment effect measured through observed indicators is often marginal or ambiguous. But most studies also contain measurements related to other latent quantities that can be used as covariates in improving the sensitivity of the intervention assessment. For example, socio-economic characteristics that pre-date the intervention are usually available. Typically, the potential covariates are also latent variables measured by a large number of observed indicators. Furthermore, the covariates\u27 relationships to the intervention-targeted response variable are often complex, and may require investigation/modeling. We propose an approach that estimates the values of latent variables and allows for an efficient and proper assessment of the intervention effect. This approach can also be useful in modeling potentially nonlinear relationships among latent variables.;Here is an abstract for the paper entitled: Improved inference procedures for true values of latent variables. The use of latent factor structure is reasonable in social, medical, business, and behavioral sciences because the theoretical constructs are often observed only indirectly through a set of observable indicators. Although estimates of standard factor scores are available, making inferences about the true value of a latent construct has not been discussed widely. In this paper, a variance estimator for the factor score estimator is derived that incorporates the additional variability due to the parameter estimation. Also, an estimated residual vector in the latent variable analysis is defined, and its properties derived. Diagnostic procedures using the factor score and residual estimates are proposed. Simulation studies and an example are given

Produto funcional de grafos :propriedades e aplicações

In this thesis, we approach the functional product of graphs, their properties, and applications. We study associativity, and show that, in the set of graphs, the relation “to have the same sequence of degrees” is an equivalence relation and the functional product is associative in the equivalence classes. With regard to the invariant connectivity, we present conditions in which the functional product of bipartite graphs generates a disconnected graph, and we generalize this result to k-partite graphs. Therefore, we improved the result reported by Lozano et al. [33], which ensures the connectivity of the functional product graph, when the factor graphs are connected. Together with Lozano and Siqueira, we show that the functional product of graphs allows to construct harmonic graphs from any regular graph. Initially, we prove that for every regular graph G and its complement G′ , there are linking applications such that the functional product graph is harmonic. After that, we show that given a regular graph G and its complement G′ , if ∆(G′ ) is even, then for any graph H such that ∆(G′ ) = ∆(H), there are linking applications such that the functional product graph is harmonic. Finally, we prove that for n and k ∈ N, if (k + 1)|n, there is a harmonic connected k-regular graph with n vertices.Nesta tese, abordamos o produto funcional de grafos, suas propriedades e aplicações. Estudamos a associatividade e mostramos que, no conjunto dos grafos, a relação “ter a mesma sequencia de graus” é uma relação de equivalência e o produto funcional ´e associativo nas classes de equivalência. No tocante a invariante conexidade, apresentamos condições em que o produto funcional de grafos bipartidos gera um grafo desconexo e generalizamos esse resultado para grafos k-partidos. Além disso, melhoramos o resultado apresentado por Lozano et al. [33], que garante a conexidade do grafo produto funcional, quando os grafos fatores são conexos. Em trabalho conjunto com Lozano e Siqueira, mostramos que o produto funcional de grafos permite construir grafos harmônicos, a partir de qualquer grafo regular. Inicialmente, provamos que para todo grafo regular G e seu complemento G′ , existem aplicações de ligação tais que o grafo produto funcional é harmônico. Em seguida, mostramos que dado um grafo regular G e seu complemento G′, se ∆(G′ ) é par, então para qualquer grafo H tal que ∆(G′ ) = ∆(H), existem aplicações de ligação tais que o grafo produto funcional é harmônico. Por fim, provamos que para n e k ∈ N, se (k + 1)|n, existe um grafo conexo harmônico k-regular com n vértices

Hypergraph Products

In this work, new definitions of hypergraph products are presented. The main focus is on the generalization of the commutative standard graph products: the Cartesian, the direct and the strong graph product. We will generalize these well-known graph products to products of hypergraphs and show several properties like associativity, commutativity and distributivity w.r.t. the disjoint union of hypergraphs. Moreover, we show that all defined products of simple (hyper)graphs result in a simple (hyper)graph. We will see, for what kind of product the projections into the factors are (at least weak) homomorphisms and for which products there are similar connections between the hypergraph products as there are for graphs. Last, we give a new and more constructive proof for the uniqueness of prime factorization w.r.t. the Cartesian product than in [Studia Sci. Math. Hungar. 2: 285–290 (1967)] and moreover, a product relation according to such a decomposition. That might help to find efficient algorithms for the decomposition of hypergraphs w.r.t. the Cartesian product

A system-theoretic approach to multi-agent models

A system-theoretic model for cooperative settings is presented that unifies and ex- tends the models of classical cooperative games and coalition formation processes and their generalizations. The model is based on the notions of system, state and transi- tion graph. The latter describes changes of a system over time in terms of actions governed by individuals or groups of individuals. Contrary to classic models, the pre- sented model is not restricted to acyclic settings and allows the transition graph to have cycles. Time-dependent solutions to allocation problems are proposed and discussed. In par- ticular, Weber’s theory of randomized values is generalized as well as the notion of semi-values. Convergence assertions are made in some cases, and the concept of the Cesàro value of an allocation mechanism is introduced in order to achieve convergence for a wide range of allocation mechanisms. Quantum allocation mechanisms are de- fined, which are induced by quantum random walks on the transition graph and it is shown that they satisfy certain fairness criteria. A concept for Weber sets and two dif- ferent concepts of cores are proposed in the acyclic case, and it is shown under some mild assumptions that both cores are subsets of the Weber set. Moreover, the model of non-cooperative games in extensive form is generalized such that the presented model achieves a mutual framework for cooperative and non-co- operative games. A coherency to welfare economics is made and to each allocation mechanism a social welfare function is proposed