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Indecomposability and signed domination in graphs
The area of indecomposability has been studied for over a decade. Indecomposability refers to graphs that have only trivial intervals. An interval is a set of vertices such that all vertices in this set have the same connections with respect to vertices outside the set. Trivial intervals are defined to be the empty set, the complete set, and any singleton. This area of research has not only helped graph theory to have a deeper insight into the structure of graphs, but it has also been useful for different applications. Decomposable graphs have a natural use in computer networks. Thus, the problems related to indecomposable graphs are not only interesting from a graph theoretic point of view, but also for use in the real-world. ^ The study of signed domination is a newer area of research compared to indecomposability. Signed domination refers to the process of labeling vertices with a -1 or +1 according to certain properties. By studying signed domination problems, we gain a better insight to the familiar problem of domination in a graph. ^ This dissertation consists of two parts. The first part is comprised of Chapter 4. In Chapter 4, we study the structural properties of indecomposable graphs as well as indecomposable subgraphs. The results deal with how to characterize and recognize indecomposable graphs according to some other object. For the first section, we characterize indecomposable graphs that are minimal for one vertex. In the second section, we use the structural properties to create k-covering graphs for k > 0. The third section deals with partially critical indecomposable graphs, which are critical according to some induced subgraph. The final section deals with extending the results of partially critical indecomposable graphs to directed graphs. ^ The second part of the dissertation is Chapter 5. In the second part, we review the signed domination number for several classes of graphs. We then present two algorithms for computing the signed dominating function for N-ary trees and interval graphs. ^
Breiner, Andrew Charles, "Indecomposability and signed domination in graphs" (2006). ETD collection for University of Nebraska - Lincoln. AAI3216432.