For a graph G, a set S is a dominating set if every vertex in V-S has a neighbor in S. A vertex contained in some minimum dominating set is called good; otherwise it is bad. A graph G has g(G) good vertices and b(G) bad vertices. The relationship between the order of G and g(G) assigns the graph to one of four classes.
Our results include a method of classifying caterpillars. Further, we develop realizability conditions for a graph G given a triple of nonnegative integers representing the domination number of γ(G), g(G), and b(G), respectively, and provide constructions of graphs meeting those conditions. We define the goodness index of a vertex v in a graph G as the ratio of distinct γ(G)-sets containing v to the total number of γ(G)-sets, and provide formulas that yield the goodness index of any vertex in a given path.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Jackson, Eugenie Marie"],"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":2001,"date_issued":"2001-05-01T07:00:00Z","date_published":"2001-05-01T07:00:00Z","updated_at":"2026-07-24T02:19:07Z","subjects":["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"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/132","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Jackson, Eugenie Marie"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["1900-01-01T08:00:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2001-05-01T07:00:00Z"]},{"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":["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"]}]},{"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/1182/viewcontent/jacksone.pdf","https://dc.etsu.edu/etd/132"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["For a graph G, a set S is a dominating set if every vertex in V-S has a neighbor in S. A vertex contained in some minimum dominating set is called good; otherwise it is bad. A graph G has g(G) good vertices and b(G) bad vertices. The relationship between the order of G and g(G) assigns the graph to one of four classes.
Our results include a method of classifying caterpillars. Further, we develop realizability conditions for a graph G given a triple of nonnegative integers representing the domination number of γ(G), g(G), and b(G), respectively, and provide constructions of graphs meeting those conditions. We define the goodness index of a vertex v in a graph G as the ratio of distinct γ(G)-sets containing v to the total number of γ(G)-sets, and provide formulas that yield the goodness index of any vertex in a given path.
"]},{"key":"dc:title","label":"Title","values":["Explorations in the Classification of Vertices as Good or Bad."]}]}],"canonical_facts":{"dc:creator":["Jackson, Eugenie Marie"],"dc:date.available":["1900-01-01T08:00:00Z"],"dc:date.issued":["2001-05-01T07:00:00Z"],"dc:description.abstract":["For a graph G, a set S is a dominating set if every vertex in V-S has a neighbor in S. A vertex contained in some minimum dominating set is called good; otherwise it is bad. A graph G has g(G) good vertices and b(G) bad vertices. The relationship between the order of G and g(G) assigns the graph to one of four classes.
Our results include a method of classifying caterpillars. Further, we develop realizability conditions for a graph G given a triple of nonnegative integers representing the domination number of γ(G), g(G), and b(G), respectively, and provide constructions of graphs meeting those conditions. We define the goodness index of a vertex v in a graph G as the ratio of distinct γ(G)-sets containing v to the total number of γ(G)-sets, and provide formulas that yield the goodness index of any vertex in a given path.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/1182/viewcontent/jacksone.pdf","https://dc.etsu.edu/etd/132"],"dc:rights":["Copyright by the authors."],"dc:subject":["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"],"dc:title":["Explorations in the Classification of Vertices as Good or Bad."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - unrestricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:19:07Z"}