Browsing by Author "Benecke, Stephen"

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    Higher order domination of graphs
    (Stellenbosch : University of Stellenbosch, 2004-12) Benecke, Stephen; Van Vuuren, J. H.; Grobler, P. J. P.; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Motivation for the study of protection strategies for graphs is rooted in antiquity and has evolved as a subdiscipline of graph theory since the early 1990s. Using, as a point of departure, the notions of weak Roman domination and secure domination (where protection of a graph is required against a single attack) an initial framework for higher order domination was introduced in 2002 (allowing for the protection of a graph against an arbitrary finite, or even infinite, number of attacks). In this thesis, the theory of higher order domination in graphs is broadened yet further to include the possibility of an arbitrary number of guards being stationed at a vertex. The thesis firstly provides a comprehensive survey of the combinatorial literature on Roman domination, weak Roman domination, secure domination and other higher order domination strategies, with a view to summarise the state of the art in the theory of higher order graph domination as at the start of 2004. Secondly, a generalised framework for higher order domination is introduced in two parts: the first catering for the protection of a graph against a finite number of consecutive attacks, and the second concerning the perpetual security of a graph (protection of the graph against an infinite number of consecutive attacks). Two types of higher order domination are distinguished: smart domination (requiring the existence of a protection strategy for any sequence of consecutive attacks of a pre–specified length, but leaving it up to a strategist to uncover such a guard movement strategy for a particular instance of the attack sequence), and foolproof domination (requiring that any possible guard movement strategy be a successful protection strategy for the graph in question). Properties of these higher order domination parameters are examined—first by investigating the application of known higher order domination results from the literature, and secondly by obtaining new results, thereby hopefully improving current understanding of these domination parameters. Thirdly, the thesis contributes by (i) establishing higher order domination parameter values for some special graph classes not previously considered (such as complete multipartite graphs, wheels, caterpillars and spiders), by (ii) summarising parameter values for special graph classes previously established (such as those for paths, cycles and selected cartesian products), and by (iii) improving higher order domination parameter bounds previously obtained (in the case of the cartesian product of two cycles). Finally, a clear indication of unresolved problems in higher order graph domination is provided in the conclusion to this thesis, together with some suggestions as to possibly desirable future generalisations of the theory.