Uwe Leck

On Orthogonal Double Covers of Graphs

An orthogonal double cover (ODC) is a collection of n spanning subgraphs(pages) of the complete graph Kn such that they cover every edge of the completegraph twice and the intersection of any two of them contains exactly one edge. If all the pages are isomorphic tosome graph G, we speak of an ODC by G. ODCs have been studied for almost 25 years, and existenceresults have been derived for many graph classes. We present an overview of the current state of research alongwith some new results and generalizations. As will be obvious, progress made in the last 10 years is in many waysrelated to the work of Ron Mullin. So it is natural and with pleasure that we dedicate this article to Ron, on theoccasion of his 65th birthday.

On Codd Families of Keys over Incomplete Relations

Keys allow a database management system to uniquely identify tuples in a database. Consequently, the class of keys is of great significance for almost all data processing tasks. In the relational model of data, keys have received considerable interest and are well understood. However, for efficient means of data processing most commercial relational database systems deviate from the relational model. For example, tuples may contain only partial information in the sense that they contain so-called null values to represent incomplete information. Codds principle of entity integrity says that every tuple of every relation must not contain a null value on any attribute of the primary key. Therefore, a key over partial relations enforces both uniqueness and totality of tuples on the attributes of the key. On the basis of these two requirements, we study the resulting class of keys over relations that permit occurrences of Zaniolos null value ‘no-information’. We show that the interaction of this class of keys is different from the interaction of the class of keys over total relations. We establish a finite ground axiomatization, and an algorithm for deciding the associated implication problem in linear time. Further, we characterize Armstrong relations for an arbitrarily given sets of keys; that is, we give a sufficient and necessary condition for a partial relation to satisfy a key precisely when it is implied by a given set of keys. We also establish an algorithm that computes an Armstrong relation for an arbitrarily given set of keys. While the problem of finding an Armstrong relation for a given key set is precisely exponential in general, our algorithm returns an Armstrong relation whose size is at most quadratic in the size of a minimal Armstrong relation. Finally, we settle various questions related to the maximal size of a family of non-redundant key sets. Our results help to bridge the gap between the existing theory of database constraints and database practice.

Logical Foundations of Possibilistic Keys

Possibility theory is applied to introduce and reason about the fundamental notion of a key for uncertain data. Uncertainty is modeled qualitatively by assigning to tuples of data a degree of possibility with which they occur in a relation, and assigning to keys a degree of certainty which says to which tuples the key applies. The associated implication problem is characterized axiomatically and algorithmically. It is shown how sets of possibilistic keys can be visualized as possibilistic Armstrong relations, and how they can be discovered from given possibilistic relations. It is also shown how possibilistic keys can be used to clean dirty data by revising the belief in possibility degrees of tuples.

Possible and certain keys for SQL

Driven by the dominance of the relational model and the requirements of modern applications, we revisit the fundamental notion of a key in relational databases with NULL. In SQL, primary key columns are NOT NULL, and UNIQUE constraints guarantee uniqueness only for tuples without NULL. We investigate the notions of possible and certain keys, which are keys that hold in some or all possible worlds that originate from an SQL table, respectively. Possible keys coincide with UNIQUE, thus providing a semantics for their syntactic definition in the SQL standard. Certain keys extend primary keys to include NULL columns and can uniquely identify entities whenever feasible, while primary keys may not. In addition to basic characterization, axiomatization, discovery, and extremal combinatorics problems, we investigate the existence and construction of Armstrong tables, and describe an indexing scheme for enforcing certain keys. Our experiments show that certain keys with NULLs occur in real-world data, and related computational problems can be solved efficiently. Certain keys are therefore semantically well founded and able to meet Codd’s entity integrity rule while handling high volumes of incomplete data from different formats.

On orthogonal double covers by trees

A collection of n spanning subgraphs of the complete graph Kn is said to be an orthogonal double cover (ODC) if every edge of Kn belongs to exactly two members of and every two elements of share exactly one edge. We consider the case when all graphs in are isomorphic to some tree G and improve former results on the existence of ODCs, especially for trees G of short diameter and for trees of G on few vertices.

Orthogonal double covers of complete graphs by trees of small diameter

Abstract A collection P of n spanning subgraphs of the complete graph K n is an orthogonal double cover (ODC) of K n if every edge of K n belongs to exactly two members of P , and if every two members of P share exactly one edge. P is an ODC of K n by some graph G if all graphs in P are isomorphic to G . Gronau, Mullin, and Rosa conjecture that every tree except the path with four vertices admits an ODC of the fitting K n . They proved this to be true for trees of diameter 3 . In this paper, we show the correctness of their conjecture for some classes of trees of diameter 4 .

Constructing Armstrong tables for general cardinality constraints and not-null constraints

Integrity constraints capture relevant requirements of an application that should be satisfied by every state of the database. The theory of integrity constraints is largely a theory over relations. To make data processing more efficient, SQL permits database states to be partial bags that can accommodate incomplete and duplicate information. Integrity constraints, however, interact differently on partial bags than on the idealized special case of relations. In this current paper, we study the implication problem of the combined class of general cardinality constraints and not-null constraints on partial bags. We investigate structural properties of Armstrong tables for general cardinality constraints and not-null constraints, and prove exact conditions for their existence. For the fragment of general max-cardinality constraints, unary min-cardinality constraints and not-null constraints we show that the effort for constructing Armstrong tables is precisely exponential. For the same fragment we provide an axiomatic characterization of the implication problem. The major tool for establishing our results is the Hajnal and Szemerédi theorem on the equitable colorings of graphs.

A Class of 2-Colorable Orthogonal Double Covers of Complete Graphs by Hamiltonian Paths

Kn by a graph G is a collection ? of n spanning subgraphs of Kn, all isomorphic to G, such that any two members of ? share exactly one edge and every edge of Kn is contained in exactly two members of ?. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-hamiltonian cycle, i.e. a cycle of length n−1. It is known that the existence of an ODC of Kn by a hamiltonian path implies the existence of ODCs of K4n and K16n, respectively, by almost-hamiltonian cycles. Horton and Nonay introduced 2-colorable ODCs and showed: If for n≥3 and a prime power q≥5 there are an ODC of Kn by a hamiltonian path and a 2-colorable ODC of Kq by a hamiltonian path, then there is an ODC of Kqn by a hamiltonian path. We construct 2-colorable ODCs of Kn and K2n, respectively, by hamiltonian paths for all odd square numbers n≥9.

Optimal shadows and ideals in submatrix orders

Abstract The main result of this article is in proving a conjecture by Sali. We obtain a Kruskal–Katona-type theorem for the poset P(N;A,B) , which for a finite set N and disjoint subsets A,B⊆N is the set {F⊆N | F∩A≠∅≠F∩B} , ordered by inclusion. Such posets are known as submatrix orders. As an application we give a solution to the problem of finding an ideal of given size and maximum weight in submatrix orders and in their duals.

A Simple Proof of the Karakhanyan-Riordan Theorem on the Even Discrete Torus

We present a simple proof of the result published by Karakhanyan [Doklady AN Arm. SSR, LXXIV (1982), pp. 61-65] and Riordan [SIAM J. Discrete Math., 11 (1998), pp. 110-127] concerning the vertex-isoperimetric problem on the