Explorations in the Classification of Vertices as Good or Bad.

<p>For a graph <em>G</em>, a set <em>S</em> is a dominating set if every vertex in <em>V</em>-<em>S</em> has a neighbor in <em>S</em>. A vertex contained in some minimum dominating set is called good; otherwise it is bad. A graph <em>G</em> has <em>g</em>(<em>G</em>) good vertices and <em>b</em>(<em>G</em>) bad vertices. The relationship between the order of <em>G</em> and <em>g</em>(<em>G</em>) assigns the graph to one of four classes.</p> <p>Our results include a method of classifying caterpillars. Further, we develop realizability conditions for a graph <em>G</em> given a triple of nonnegative integers representing the domination number of <em>γ</em>(<em>G</em>), <em>g</em>(<em>G</em>), and <em>b</em>(<em>G</em>), respectively, and provide constructions of graphs meeting those conditions. We define the goodness index of a vertex <em>v</em> in a graph <em>G</em> as the ratio of distinct <em>γ</em>(<em>G</em>)-sets containing <em>v</em> to the total number of <em>γ</em>(<em>G</em>)-sets, and provide formulas that yield the goodness index of any vertex in a given path.</p>

Name thesis:degree_name

MS (Master of Science)

Level thesis:degree_level

Thesis - unrestricted

Discipline thesis:degree_discipline

Mathematical Sciences

Year dc:date.issued

2001

Author dc:creator

• Jackson, Eugenie Marie

• graph theory
• bad vertices
• domination
• domination-fair graphs
• realizability
• caterpillars
• paths
• domination-commendable graphs
• domination-excellent graphs
• goodness index
• good vertices
• domination-poor graphs
• Physical Sciences and Mathematics

Statement dc:rights

• Copyright by the authors.

Repository record dc:identifier

https://dc.etsu.edu/etd/132

OAI identifier oai:identifier

oai:dc.etsu.edu:etd-1182