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, gammat, is the cardinality of a smallest total dominating set in a graph. We investigate gammat-excellent, gammat-commendable, gammat-fair, and gammat-poor graphs.
","abstract_html":"<p>Fricke, Haynes, Hedetniemi, Hedetniemi, and Laskar introduced the following concept. For a graph <em>G</em> = (<em>V</em>,<em>E</em>), let <em>rho</em> denote a property of interest concerning sets of vertices. A vertex <em>u</em> is <em>rho-good</em> if <em>u</em> is contained in a {minimum, maximum} <em>rho-set</em> in <em>G</em> and <em>rho-bad</em> if <em>u</em> is not contained in a <em>rho-set</em>. Let <em>g</em> denote the number of <em>rho-good</em> vertices and <em>b</em> denote the number of <em>rho-bad</em> vertices. A graph <em>G</em> is called <em>rho-excellent</em> if every vertex in <em>V</em> is <em>rho</em>-good, <em>rho-commendable</em> if <em>g</em> > <em>b</em> > 0, <em>rho-fair</em> if <em>g</em> = <em>b</em>, and <em>rho-poor</em> if <em>g</em> < <em>b</em>. In this thesis the property of interest is total domination. The total domination number, <em>gamma<sub>t</sub></em>, is the cardinality of a smallest total dominating set in a graph. We investigate <em>gamma<sub>t</sub></em>-excellent, <em>gamma<sub>t</sub></em>-commendable, <em>gamma<sub>t</sub></em>-fair, and <em>gamma<sub>t</sub></em>-poor graphs.</p>","abstract_has_math":false,"creators":["Dautermann, Robert Elmer, III"],"institution":null,"degree_name":"MS (Master of Science)","degree_level":"Thesis - unrestricted","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2000,"date_issued":"2000-05-01T07:00:00Z","date_published":"2000-05-01T07:00:00Z","updated_at":"2026-07-24T02:18:50Z","subjects":["domination","total domination","graph theory","mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/5","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Dautermann, Robert Elmer, III"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2000-01-01T08:00:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2000-05-01T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - unrestricted"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS (Master of Science)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["domination","total domination","graph theory","mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Copyright by the authors."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.etsu.edu/context/etd/article/1037/viewcontent/RobThesis.pdf","https://dc.etsu.edu/etd/5"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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, gammat, is the cardinality of a smallest total dominating set in a graph. We investigate gammat-excellent, gammat-commendable, gammat-fair, and gammat-poor graphs.
"]},{"key":"dc:title","label":"Title","values":["Vertices in Total Dominating Sets."]}]}],"canonical_facts":{"dc:creator":["Dautermann, Robert Elmer, III"],"dc:date.available":["2000-01-01T08:00:00Z"],"dc:date.issued":["2000-05-01T07:00:00Z"],"dc:description.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, gammat, is the cardinality of a smallest total dominating set in a graph. We investigate gammat-excellent, gammat-commendable, gammat-fair, and gammat-poor graphs.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/1037/viewcontent/RobThesis.pdf","https://dc.etsu.edu/etd/5"],"dc:rights":["Copyright by the authors."],"dc:subject":["domination","total domination","graph theory","mathematics","Physical Sciences and Mathematics"],"dc:title":["Vertices in Total Dominating Sets."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - unrestricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:18:50Z"}