{"id":{"repo_id":"gatech","oai_identifier":"oai:repository.gatech.edu:1853/58301"},"canonical_url":"https://search.dev.ndltd.org/etd/gatech/oai:repository.gatech.edu:1853/58301","repository":{"repo_id":"gatech","name":"Georgia Tech","base_url":"https://repository.gatech.edu/server/oai/request"},"display":{"title":"Special TK5 in graphs containing K4-","abstract":"Given a graph K, TK is used to denote a subdivision of K, which is a graph obtained from K by substituting some edges for paths. The well-known Kelmans-Seymour conjecture states that every nonplanar 5-connected graph contains TK5 . Ma and Yu proved the conjecture for graphs containing K4-. In this dissertation, we strengthen their result in two ways. The results will be useful for completely resolving the Kelmans-Seymour conjecture. Let G be a 5-connected nonplanar graph and let x1, x2, y1, y2 in V(G) be distinct, such that G[{x1, x2, y1, y2}] is isomorphic to K4- and y1y2 is not in E(G). We show that one of the following holds: G - y2 contains K4-, or G contains a TK5 in which y2 is not a branch vertex, or G has a special 5-separation, or for any distinct w1, w2, w3 in N(y2) - {x1, x2}, G - {y2v : v not in {x1, x2, w1, w2, w3}} contains TK5. We show that one of the following holds: G - x1 contains K4-, or G contains a TK5 in which x1 is not a branch vertex, or G contains a K4- in which x1 is of degree 2, or {x2, y1, y2} may be chosen so that for any distinct z0, z1 in N(x1) - {x2, y1, y2}, G - {x1v : v not in {z0, z1, x2, y1, y2}} contains TK5.","abstract_html":"Given a graph K, TK is used to denote a subdivision of K, which is a graph obtained from K by substituting some edges for paths. The well-known Kelmans-Seymour conjecture states that every nonplanar 5-connected graph contains TK5 . Ma and Yu proved the conjecture for graphs containing K4-. In this dissertation, we strengthen their result in two ways. The results will be useful for completely resolving the Kelmans-Seymour conjecture. Let G be a 5-connected nonplanar graph and let x1, x2, y1, y2 in V(G) be distinct, such that G[{x1, x2, y1, y2}] is isomorphic to K4- and y1y2 is not in E(G). We show that one of the following holds: G - y2 contains K4-, or G contains a TK5 in which y2 is not a branch vertex, or G has a special 5-separation, or for any distinct w1, w2, w3 in N(y2) - {x1, x2}, G - {y2v : v not in {x1, x2, w1, w2, w3}} contains TK5. We show that one of the following holds: G - x1 contains K4-, or G contains a TK5 in which x1 is not a branch vertex, or G contains a K4- in which x1 is of degree 2, or {x2, y1, y2} may be chosen so that for any distinct z0, z1 in N(x1) - {x2, y1, y2}, G - {x1v : v not in {z0, z1, x2, y1, y2}} contains TK5.","abstract_has_math":false,"creators":["He, Dawei"],"institution":"Georgia Institute of Technology","degree_name":null,"degree_level":"Doctoral","degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Yu, Xingxing"],"committee_chairs":[],"committee_members":["Thomas, Robin","Trotter, William","Blekhermann, Greg","Li, Geoffrey Ye"],"year":2017,"date_issued":"2017-04-05","date_published":"2017-04-05","updated_at":"2026-07-27T19:50:35Z","subjects":["Kelmans-Seymour conjecture","Subdivision of K5","K4-","Branch vertex"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1853/58301","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Yu, Xingxing"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Thomas, Robin","Trotter, William","Blekhermann, Greg","Li, Geoffrey Ye"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["He, Dawei"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-06-07T17:47:36Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-06-07T17:47:36Z"]},{"key":"dc:date.issued","label":"Date","values":["2017-04-05"]},{"key":"dc:publisher","label":"Institution","values":["Georgia Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Kelmans-Seymour conjecture","Subdivision of K5","K4-","Branch vertex"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1853/58301"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Given a graph K, TK is used to denote a subdivision of K, which is a graph obtained from K by substituting some edges for paths. The well-known Kelmans-Seymour conjecture states that every nonplanar 5-connected graph contains TK5 . Ma and Yu proved the conjecture for graphs containing K4-. In this dissertation, we strengthen their result in two ways. The results will be useful for completely resolving the Kelmans-Seymour conjecture. Let G be a 5-connected nonplanar graph and let x1, x2, y1, y2 in V(G) be distinct, such that G[{x1, x2, y1, y2}] is isomorphic to K4- and y1y2 is not in E(G). We show that one of the following holds: G - y2 contains K4-, or G contains a TK5 in which y2 is not a branch vertex, or G has a special 5-separation, or for any distinct w1, w2, w3 in N(y2) - {x1, x2}, G - {y2v : v not in {x1, x2, w1, w2, w3}} contains TK5. We show that one of the following holds: G - x1 contains K4-, or G contains a TK5 in which x1 is not a branch vertex, or G contains a K4- in which x1 is of degree 2, or {x2, y1, y2} may be chosen so that for any distinct z0, z1 in N(x1) - {x2, y1, y2}, G - {x1v : v not in {z0, z1, x2, y1, y2}} contains TK5."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Special TK5 in graphs containing K4-"]}]}],"canonical_facts":{"dc:contributor.advisor":["Yu, Xingxing"],"dc:contributor.committeemember":["Thomas, Robin","Trotter, William","Blekhermann, Greg","Li, Geoffrey Ye"],"dc:contributor.department":["Mathematics"],"dc:creator":["He, Dawei"],"dc:date.accessioned":["2017-06-07T17:47:36Z"],"dc:date.available":["2017-06-07T17:47:36Z"],"dc:date.issued":["2017-04-05"],"dc:description.abstract":["Given a graph K, TK is used to denote a subdivision of K, which is a graph obtained from K by substituting some edges for paths. The well-known Kelmans-Seymour conjecture states that every nonplanar 5-connected graph contains TK5 . Ma and Yu proved the conjecture for graphs containing K4-. In this dissertation, we strengthen their result in two ways. The results will be useful for completely resolving the Kelmans-Seymour conjecture. Let G be a 5-connected nonplanar graph and let x1, x2, y1, y2 in V(G) be distinct, such that G[{x1, x2, y1, y2}] is isomorphic to K4- and y1y2 is not in E(G). We show that one of the following holds: G - y2 contains K4-, or G contains a TK5 in which y2 is not a branch vertex, or G has a special 5-separation, or for any distinct w1, w2, w3 in N(y2) - {x1, x2}, G - {y2v : v not in {x1, x2, w1, w2, w3}} contains TK5. We show that one of the following holds: G - x1 contains K4-, or G contains a TK5 in which x1 is not a branch vertex, or G contains a K4- in which x1 is of degree 2, or {x2, y1, y2} may be chosen so that for any distinct z0, z1 in N(x1) - {x2, y1, y2}, G - {x1v : v not in {z0, z1, x2, y1, y2}} contains TK5."],"dc:description.degree":["Ph.D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1853/58301"],"dc:language.iso":["en_US"],"dc:publisher":["Georgia Institute of Technology"],"dc:subject":["Kelmans-Seymour conjecture","Subdivision of K5","K4-","Branch vertex"],"dc:title":["Special TK5 in graphs containing K4-"],"dc:type":["Text"],"thesis:degree_level":["Doctoral"]},"updated_at":"2026-07-27T19:50:35Z"}