Locating-dominating sets and identifying codes in graphs of girth at least 5

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.Comment: 20 pages, 9 figure

Fingerprinting Codes and Separating Hash Families

The thesis examines two related combinatorial objects, namely fingerprinting codes and separating hash families. Fingerprinting codes are combinatorial objects that have been studied for more than 15 years due to their applications in digital data copyright protection and their combinatorial interest. Four well-known types of fingerprinting codes are studied in this thesis; traceability, identifiable parent property, secure frameproof and frameproof. Each type of code is named after the security properties it guarantees. However, the power of these four types of fingerprinting codes is limited by a certain condition. The first known attempt to go beyond that came out in the concept of two-level traceability codes, introduced by Anthapadmanabhan and Barg (2009). This thesis extends their work to the other three types of fingerprinting codes, so in this thesis four types of two-level fingerprinting codes are defined. In addition, the relationships between the different types of codes are studied. We propose some first explicit non-trivial constructions for two-level fingerprinting codes and provide some bounds on the size of these codes. Separating hash families were introduced by Stinson, van Trung, and Wei as a tool for creating an explicit construction for frameproof codes in 1998. In this thesis, we state a new definition of separating hash families, and mainly focus on improving previously known bounds for separating hash families in some special cases that related to fingerprinting codes. We improve upper bounds on the size of frameproof and secure frameproof codes under the language of separating hash families

Almost separating and almost secure frameproof codes over q-ary alphabets

The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-015-0060-zIn this paper we discuss some variations of the notion of separating code for alphabets of arbitrary size. We show how the original definition can be relaxed in two different ways, namely almost separating and almost secure frameproof codes, yielding two different concepts. The new definitions enable us to obtain codes of higher rate, at the expense of satisfying the separating property partially. These new definitions become useful when complete separation is only required with high probability, rather than unconditionally. We also show how the codes proposed can be used to improve the rate of existing constructions of families of fingerprinting codes.Peer ReviewedPostprint (author's final draft

Linear time Constructions of some dd-Restriction Problems

We give new linear time globally explicit constructions for perfect hash families, cover-free families and separating hash functions