On subgraphs of Cartesian product graphs and S-primeness

AbstractIn this paper we consider S-prime graphs, that is the graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs. Lamprey and Barnes characterized S-prime graphs via so-called basic S-prime graphs that form a subclass of all S-prime graphs. However, the structure of basic S-prime graphs was not known very well. In this paper we prove several characterizations of basic S-prime graphs. In particular, the structural characterization of basic S-prime graphs of connectivity 2 enables us to present several infinite families of basic S-prime graphs. Furthermore, simple S-prime graphs are introduced that form a relatively small subclass of basic S-prime graphs, and it is shown that every basic S-prime graph can be obtained from a simple S-prime graph by a sequence of certain transformations called extensions

On the Complexity of Recognizing S-composite and S-prime Graphs

S-prime graphs are graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs, i.e., whenever it is a subgraph of a nontrivial Cartesian product graph it is a subgraph of one the factors. A graph is S-composite if it is not S-prime. Although linear time recognition algorithms for determining whether a graph is prime or not with respect to the Cartesian product are known, it remained unknown if a similar result holds also for the recognition of S-prime and S-composite graphs. In this contribution the computational complexity of recognizing S-composite and S-prime graphs is considered. Klav{\v{z}}ar \emph{et al.} [\emph{Discr.\ Math.} \textbf{244}: 223-230 (2002)] proved that a graph is S-composite if and only if it admits a nontrivial path-$k$-coloring. The problem of determining whether there exists a path-$k$-coloring for a given graph is shown to be NP-complete even for $k=2$. This in turn is utilized to show that determining whether a graph is S-composite is NP-complete and thus, determining whether a graph is S-prime is CoNP-complete. Many other problems are shown to be NP-hard, using the latter results

Local Prime Factor Decomposition of Approximate Strong Product Graphs: Local Prime Factor Decompositionof Approximate Strong Product Graphs

In practice, graphs often occur as perturbed product structures, so-called approximate graph products. The practical application of the well-known prime factorization algorithms is therefore limited, since most graphs are prime, although they can have a product-like structure. This work is concerned with the strong graph product. Since strong product graphs G contain subgraphs that are itself products of subgraphs of the underlying factors of G, we follow the idea to develop local approaches that cover a graph by factorizable patches and then use this information to derive the global factors. First, we investigate the local structure of strong product graphs and introduce the backbone B(G) of a graph G and the so-called S1-condition. Both concepts play a central role for determining the prime factors of a strong product graph in a unique way. Then, we discuss several graph classes, in detail, NICE, CHIC and locally unrefined graphs. For each class we construct local, quasi-linear time prime factorization algorithms. Combining these results, we then derive a new local prime factorization algorithm for all graphs. Finally, we discuss approximate graph products. We use the new local factorization algorithm to derive a method for the recognition of approximate graph products. Furthermore, we evaluate the performance of this algorithm on a sample of approximate graph products

Bunched logics: a uniform approach

Bunched logics have found themselves to be key tools in modern computer science, in particular through the industrial-level program verification formalism Separation Logic. Despite this—and in contrast to adjacent families of logics like modal and substructural logic—there is a lack of uniform methodology in their study, leaving many evident variants uninvestigated and many open problems unresolved. In this thesis we investigate the family of bunched logics—including previously unexplored intuitionistic variants—through two uniform frameworks. The first is a system of duality theorems that relate the algebraic and Kripke-style interpretations of the logics; the second, a modular framework of tableaux calculi that are sound and complete for both the core logics themselves, as well as many classes of bunched logic model important for applications in program verification and systems modelling. In doing so we are able to resolve a number of open problems in the literature, including soundness and completeness theorems for intuitionistic variants of bunched logics, classes of Separation Logic models and layered graph models; decidability of layered graph logics; a characterisation theorem for the classes of bunched logic model definable by bunched logic formulae; and the failure of Craig interpolation for principal bunched logics. We also extend our duality theorems to the categorical structures suitable for interpreting predicate versions of the logics, in particular hyperdoctrinal structures used frequently in Separation Logic

Reasoning with Inconsistent Information

In this thesis we are concerned with developing formal and representational mechanisms for reasoning with inconsistent information. Strictly speaking there are two conceptually distinct senses in which we are interested in reasoning with inconsistent information. In one sense, we are interested in using logical deduction to draw inferences in a symbolic system. More specifically, we are interested in mechanisms that can continue to perform deduction in a reasonable manner despite the threat of inconsistencies as a direct result of errors or misrepresentations. So in this sense we are interested in inconsistency-tolerant or paraconsistent deduction. … ¶ In this thesis we adopt a novel framework to unify both logic-as-deduction and logic-as-representation approaches to reasoning with inconsistent information. …

Topological entanglement complexity of systems of polygons and walks in tubes

In this thesis, motivated by modelling polymers, the topological entanglement complexity of systems of two self-avoiding polygons (2SAPs), stretched polygons and systems of self-avoiding walks (SSAWs) in a tubular sublattice of Z3 are investigated. In particular, knotting and linking probabilities are used to measure a polygon fs selfentanglement and its entanglement with other polygons respectively. For the case of 2SAPs, it is established that the homological linking probability goes to one at least as fast as 1-O(n^(-1/2)) and that the topological linking probability goes to one exponentially rapidly as n, the size of the 2SAP, goes to infinity. For the case of stretched polygons, used to model ring polymers under the influence of an external force f, it is shown that, no matter the strength or direction of the external force, the knotting probability goes to one exponentially as n, the size of the polygon, goes to infinity. Associating a two-component link to each stretched polygon, it is also proved that the topological linking probability goes to unity exponentially fast as n → ∞. Furthermore, a set of entangled chains confined to a tube is modelled by a system of self- and mutually avoiding walks (SSAW). It is shown that there exists a positive number γ such that the probability that an SSAW of size n has entanglement complexity (EC), as defined in this thesis, greater than γn approaches one exponentially as n → ∞. It is also established that EC of an SSAW is bounded above by a linear function of its size. Using a transfer-matrix approach, the asymptotic form of the free energy for the SSAW model is also obtained and the average edge-density for span m SSAWs is proved to approach a constant as m → ∞. Hence, it is shown that EC is a ggood h measure of entanglement complexity for dense polymer systems modelled by SSAWs, in particular, because EC increases linearly with system size, as the size of the system goes to infinity