Candidate: Ralucca M. Gera

Degree of: Doctor of Philosophy

Department: Mathematics

Title: Stratification and Domination in Graphs and Digraphs

Committee: Dr. Ping Zhang, Chair
Dr. Gary Chartrand
Dr. Clifton Ealy
Dr. Allen Schwenk
Dr. Garry Johns

Date: Tuesday, November 9, 2004 3:00 p.m.-5:00 p.m.
6625 Everett Tower, Alavi Commons

Abstract: In this Theses we combine the study of domination with the study of stratification in graphs and digraphs.
Recently, stratification became part of Graph Theory with its applications in Very Large Integrated Circuit (VLSI) chips. On the other hand, domination has been extensively studied, but not from the point of view of stratification. Chartrand, Haynes, Henning and Zhang combined domination and stratification in graphs, and this Theses extends the results in their research paper. Domination in stratified graphs generalizes the standard domination, as well as other types of domination well known in graph theory. One advantage of studying domination in stratified graphs versus common domination is that the first is always defined. For a particular graph F, we study domination in stratified graphs, called F-domination. The F-domination number, associated to this type of domination, is defined and determined for some well-known classes of graphs. Bounds for the F-domination number of graphs are established in terms of its order, clique number, diameter, girth, and maximum degree. Characterizations for graphs of order n with F-domination number n or 1 are presented.
We also compare the domination number to other related common domination parameters, such as domination, open domination, restrained domination and 2-step domination. Relationships between F-domination and these four domination parameters are presented, as well as complete realization results of domination, F-domination, and open domination, as a triple.
Domination in stratified digraphs is also studied, and in particular in regular digraphs. Realization results and bounds are presented.
This research presents a broader view to domination, bringing out the opportunity for deeper research of the topic which might change the current view of domination in graphs and digraphs.

Mathematical abstract available at: http://homepages.wmich.edu/~s8muntea/abstract.pdf

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