{"id":{"repo_id":"lsu-thes","oai_identifier":"oai:repository.lsu.edu:gradschool_dissertations-2367"},"canonical_url":"https://search.dev.ndltd.org/etd/lsu-thes/oai:repository.lsu.edu:gradschool_dissertations-2367","repository":{"repo_id":"lsu-thes","name":"Lousiana State University","base_url":"https://repository.lsu.edu/do/oai/"},"display":{"title":"Excluding a Weakly 4-connected Minor","abstract":"A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \\leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \\leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs.","abstract_html":"A 3-connected graph $G$ is called weakly 4-connected if min <span class=\"etd-inline-math\">(|E(G<sub>1</sub>)|, |E(G<sub>2</sub>)|) \\leq 4</span> holds for all 3-separations <span class=\"etd-inline-math\">(G<sub>1</sub>,G<sub>2</sub>)</span> of $G$. A 3-connected graph $G$ is called quasi 4-connected if min <span class=\"etd-inline-math\">(|V(G<sub>1</sub>)|, |V(G<sub>2</sub>)|) \\leq 4</span>. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs.","abstract_has_math":true,"creators":["D'souza, Kimberly Sevin"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-01-01T08:00:00Z","date_published":"2016-01-01T08:00:00Z","updated_at":"2026-07-24T02:59:22Z","subjects":["graph theory","graph minors","graph decomposition","Pyramid graph"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-04042016-220803","https://repository.lsu.edu/gradschool_dissertations/1368"],"render_values":[{"text":"etd-04042016-220803","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/1368","href":"https://repository.lsu.edu/gradschool_dissertations/1368","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.1368","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["D'souza, Kimberly Sevin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2016-03-14"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:11:54Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["graph theory","graph minors","graph decomposition","Pyramid graph"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-04042016-220803","10.31390/gradschool_dissertations.1368","https://repository.lsu.edu/gradschool_dissertations/1368"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \\leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \\leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs."]},{"key":"dc:title","label":"Title","values":["Excluding a Weakly 4-connected Minor"]}]}],"canonical_facts":{"dc:creator":["D'souza, Kimberly Sevin"],"dc:date":["2016-03-14"],"dc:date.available":["2022-05-12T23:11:54Z"],"dc:description.abstract":["A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \\leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \\leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs."],"dc:identifier":["etd-04042016-220803","10.31390/gradschool_dissertations.1368","https://repository.lsu.edu/gradschool_dissertations/1368"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["graph theory","graph minors","graph decomposition","Pyramid graph"],"dc:title":["Excluding a Weakly 4-connected Minor"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:59:22Z"}