The probabilistic method for upper bounds in domination theory

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and 3 are devoted to k-domination, k-tuple domination, k-total domination, α-domination and α-rate domination in graphs. A review of wellknown results is given, and the new results are presented. These new upper bounds generalize two classical bounds for the single domination number and also improve a number of known bounds for the k-domination and k-tuple domination numbers. Effective randomized algorithms are given for finding k-dominating, ktuple dominating, α-dominating and α-rate dominating sets, whose expected sizes satisfy the above upper bounds. These algorithms follow from the probabilistic constructions used to prove the corresponding upper bounds. Similar research is carried out for the the global and Roman domination parameters in Chapter 4. New upper bounds for the global domination and Roman domination numbers are presented, and it is proved that these results are asymptotically best possible. Moreover, the upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs are given, and it is shown that, for almost all graphs, the restrained domination number is equal to the domination number, and the total restrained domination number is equal to the total domination number. Signed domination is another domination parameter studied in Chapter 5. This concept is closely related to combinatorial discrepancy theory. New upper and lower bounds for the signed domination number are presented. These new bounds improve a number of known results. Moreover, Füredi–Mubayi’s conjecture is rectified.