#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

#### Date of Award

5-2000

#### Committee Chair or Co-Chairs

Teresa W. Haynes

#### Committee Members

Debra J. Knisley, James Boland

#### Abstract

Fricke, Haynes, Hedetniemi, Hedetniemi, and Laskar introduced the following concept. For a graph *G* = (*V*,*E*), let *rho* denote a property of interest concerning sets of vertices. A vertex *u* is *rho-good* if *u* is contained in a {minimum, maximum} *rho-set* in *G* and *rho-bad* if *u* is not contained in a *rho-set*. Let *g* denote the number of *rho-good* vertices and *b* denote the number of *rho-bad* vertices. A graph *G* is called *rho-excellent* if every vertex in *V* is *rho*-good, *rho-commendable* if *g* > *b* > 0, *rho-fair* if *g* = *b*, and *rho-poor* if *g* < *b*. In this thesis the property of interest is total domination. The total domination number, *gamma _{t}*, is the cardinality of a smallest total dominating set in a graph. We investigate

*gamma*-excellent,

_{t}*gamma*-commendable,

_{t}*gamma*-fair, and

_{t}*gamma*-poor graphs.

_{t}#### Document Type

Thesis - Open Access

#### Recommended Citation

Dautermann, Robert Elmer III, "Vertices in Total Dominating Sets." (2000). *Electronic Theses and Dissertations.* Paper 5. http://dc.etsu.edu/etd/5

#### Copyright

Copyright by the authors.