Every graph G = (V, E) has a dominating set S ⊆ V(G) such that any vertex not in S is adjacent to a vertex in S. We define a paired-dominating set S to be a dominating set S = {v1, v2,..., v2t-1, v2t} where M = {v1v2, v3v4, ..., v2t-1v2t} is a perfect matching in 〈S〉, the subgraph induced by S. The domination number of a graph G is the smallest cardinality of any dominating set of G, and the paired-domination number is the smallest cardinality of any paired-dominating set. Determining the domination number for grid graphs is a well-known open problem in graph theory. Not surprisingly, determining the paired-domination number for grid graphs is also a difficult problem. In this thesis, we survey past research in domination, paired-domination and grid graphs to obtain background for our study of paired-domination in grid graphs. We determine the paired-domination number for grid graphs Gr,c where r ∈ {2,3}, for infinite dimensional grid graphs, and for the complement of a grid graph.
","abstract_html":"<p>Every graph <em>G</em> = (<em>V</em>, <em>E</em>) has a dominating set <em>S</em> ⊆ <em>V</em>(<em>G</em>) such that any vertex not in <em>S</em> is adjacent to a vertex in <em>S</em>. We define a paired-dominating set <em>S</em> to be a dominating set <em>S</em> = {<em>v</em><sub>1</sub>, <em>v</em><sub>2</sub>,..., <em>v</em><sub>2<em>t</em>-1</sub>, <em>v</em><sub>2<em>t</em></sub>} where <em>M</em> = {<em>v</em><sub>1</sub><em>v</em><sub>2</sub>, <em>v</em><sub>3</sub><em>v</em><sub>4</sub>, ..., <em>v</em><sub>2<em>t</em>-1</sub><em>v</em><sub>2<em>t</em></sub>} is a perfect matching in 〈<em>S</em>〉, the subgraph induced by <em>S</em>. The domination number of a graph <em>G</em> is the smallest cardinality of any dominating set of <em>G</em>, and the paired-domination number is the smallest cardinality of any paired-dominating set. Determining the domination number for grid graphs is a well-known open problem in graph theory. Not surprisingly, determining the paired-domination number for grid graphs is also a difficult problem. In this thesis, we survey past research in domination, paired-domination and grid graphs to obtain background for our study of paired-domination in grid graphs. We determine the paired-domination number for grid graphs <em>G<sub>r</sub></em>,<em>c</em> where <em>r</em> ∈ {2,3}, for infinite dimensional grid graphs, and for the complement of a grid graph.</p>","abstract_has_math":false,"creators":["Proffitt, Kenneth Eugene"],"institution":null,"degree_name":"MS (Master of Science)","degree_level":"Thesis - restricted","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":["paired-domination","grid graphs","domination","Physical Sciences and Mathematics"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/131","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Proffitt, Kenneth Eugene"]}]},{"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 - restricted"]},{"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":["paired-domination","grid graphs","domination","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/1181/viewcontent/proffittk.pdf","https://dc.etsu.edu/etd/131"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Every graph G = (V, E) has a dominating set S ⊆ V(G) such that any vertex not in S is adjacent to a vertex in S. We define a paired-dominating set S to be a dominating set S = {v1, v2,..., v2t-1, v2t} where M = {v1v2, v3v4, ..., v2t-1v2t} is a perfect matching in 〈S〉, the subgraph induced by S. The domination number of a graph G is the smallest cardinality of any dominating set of G, and the paired-domination number is the smallest cardinality of any paired-dominating set. Determining the domination number for grid graphs is a well-known open problem in graph theory. Not surprisingly, determining the paired-domination number for grid graphs is also a difficult problem. In this thesis, we survey past research in domination, paired-domination and grid graphs to obtain background for our study of paired-domination in grid graphs. We determine the paired-domination number for grid graphs Gr,c where r ∈ {2,3}, for infinite dimensional grid graphs, and for the complement of a grid graph.
"]},{"key":"dc:title","label":"Title","values":["Paired-Domination in Grid Graphs."]}]}],"canonical_facts":{"dc:creator":["Proffitt, Kenneth Eugene"],"dc:date.available":["1900-01-01T08:00:00Z"],"dc:date.issued":["2001-05-01T07:00:00Z"],"dc:description.abstract":["Every graph G = (V, E) has a dominating set S ⊆ V(G) such that any vertex not in S is adjacent to a vertex in S. We define a paired-dominating set S to be a dominating set S = {v1, v2,..., v2t-1, v2t} where M = {v1v2, v3v4, ..., v2t-1v2t} is a perfect matching in 〈S〉, the subgraph induced by S. The domination number of a graph G is the smallest cardinality of any dominating set of G, and the paired-domination number is the smallest cardinality of any paired-dominating set. Determining the domination number for grid graphs is a well-known open problem in graph theory. Not surprisingly, determining the paired-domination number for grid graphs is also a difficult problem. In this thesis, we survey past research in domination, paired-domination and grid graphs to obtain background for our study of paired-domination in grid graphs. We determine the paired-domination number for grid graphs Gr,c where r ∈ {2,3}, for infinite dimensional grid graphs, and for the complement of a grid graph.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/1181/viewcontent/proffittk.pdf","https://dc.etsu.edu/etd/131"],"dc:rights":["Copyright by the authors."],"dc:subject":["paired-domination","grid graphs","domination","Physical Sciences and Mathematics"],"dc:title":["Paired-Domination in Grid Graphs."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - restricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:19:07Z"}