{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/88719"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/88719","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Circularity of graphs","abstract":"Let G be a finite connected graph. The circularity of G has been previously defined as σ(G) = max{r ε N| G has a circular covering of r elements, each element being a closed, connected subset of G containing at least one vertex of G}. This definition is known to be equivalent to the combinatorial description, σ(G) = max{r ε N| there is an admissible map f:V(G)→A(r)}. In this thesis, co-admissible maps are introduced and the co-circularity of a graph, G, is defined as η(G) = max{n ε N| there is a co-admissible map g:V(G)→Z<sub>n</sub>}. It is shown that σ(G) = 2η(G) or 2η(G) + 1. It is also shown that if G is a graph and g:V(G)→Z<sub>n</sub> is a co-admissible map, then G contains a cycle, J, called a co-admissible cycle, for which g:V(J)→Z<sub>n</sub> is also co-admissible. Necessary and sufficient conditions are given for extending a co-admissible map on a cycle of a graph to the entire graph. If G is a graph with σ(G) = r, it is shown that any suspended (v,w)-path P in G induces, under any admissible map f:V(G)→A(r), either at most four elements of Z<sub>r</sub> or every vertex of P with valency two induces exactly two elements of Z<sub>r</sub> not induced by any other vertex of G. Finally it is shown that if G is a planar graph and if g:V(G)→Z<sub>n</sub> is a co-admissible map, then any planar representation of G has exactly two faces bounded by co-admissible cycles.","abstract_html":"Let G be a finite connected graph. The circularity of G has been previously defined as σ(G) = max{r ε N| G has a circular covering of r elements, each element being a closed, connected subset of G containing at least one vertex of G}. This definition is known to be equivalent to the combinatorial description, σ(G) = max{r ε N| there is an admissible map f:V(G)→A(r)}. In this thesis, co-admissible maps are introduced and the co-circularity of a graph, G, is defined as η(G) = max{n ε N| there is a co-admissible map g:V(G)→Z&lt;sub&gt;n&lt;/sub&gt;}. It is shown that σ(G) = 2η(G) or 2η(G) + 1. It is also shown that if G is a graph and g:V(G)→Z&lt;sub&gt;n&lt;/sub&gt; is a co-admissible map, then G contains a cycle, J, called a co-admissible cycle, for which g:V(J)→Z&lt;sub&gt;n&lt;/sub&gt; is also co-admissible. Necessary and sufficient conditions are given for extending a co-admissible map on a cycle of a graph to the entire graph. If G is a graph with σ(G) = r, it is shown that any suspended (v,w)-path P in G induces, under any admissible map f:V(G)→A(r), either at most four elements of Z&lt;sub&gt;r&lt;/sub&gt; or every vertex of P with valency two induces exactly two elements of Z&lt;sub&gt;r&lt;/sub&gt; not induced by any other vertex of G. Finally it is shown that if G is a planar graph and if g:V(G)→Z&lt;sub&gt;n&lt;/sub&gt; is a co-admissible map, then any planar representation of G has exactly two faces bounded by co-admissible cycles.","abstract_has_math":false,"creators":["Blum, Dorothee Jane"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1982,"date_issued":"1982","date_published":"1982","updated_at":"2026-07-22T22:18:46Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/88719","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Blum, Dorothee Jane"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-03-26T19:53:31Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-03-26T19:53:31Z"]},{"key":"dc:date.issued","label":"Date","values":["1982"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/88719"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let G be a finite connected graph. The circularity of G has been previously defined as σ(G) = max{r ε N| G has a circular covering of r elements, each element being a closed, connected subset of G containing at least one vertex of G}. This definition is known to be equivalent to the combinatorial description, σ(G) = max{r ε N| there is an admissible map f:V(G)→A(r)}. In this thesis, co-admissible maps are introduced and the co-circularity of a graph, G, is defined as η(G) = max{n ε N| there is a co-admissible map g:V(G)→Z<sub>n</sub>}. It is shown that σ(G) = 2η(G) or 2η(G) + 1. It is also shown that if G is a graph and g:V(G)→Z<sub>n</sub> is a co-admissible map, then G contains a cycle, J, called a co-admissible cycle, for which g:V(J)→Z<sub>n</sub> is also co-admissible. Necessary and sufficient conditions are given for extending a co-admissible map on a cycle of a graph to the entire graph. If G is a graph with σ(G) = r, it is shown that any suspended (v,w)-path P in G induces, under any admissible map f:V(G)→A(r), either at most four elements of Z<sub>r</sub> or every vertex of P with valency two induces exactly two elements of Z<sub>r</sub> not induced by any other vertex of G. Finally it is shown that if G is a planar graph and if g:V(G)→Z<sub>n</sub> is a co-admissible map, then any planar representation of G has exactly two faces bounded by co-admissible cycles."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Circularity of graphs"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Blum, Dorothee Jane"],"dc:date.accessioned":["2019-03-26T19:53:31Z"],"dc:date.available":["2019-03-26T19:53:31Z"],"dc:date.issued":["1982"],"dc:description.abstract":["Let G be a finite connected graph. The circularity of G has been previously defined as σ(G) = max{r ε N| G has a circular covering of r elements, each element being a closed, connected subset of G containing at least one vertex of G}. This definition is known to be equivalent to the combinatorial description, σ(G) = max{r ε N| there is an admissible map f:V(G)→A(r)}. In this thesis, co-admissible maps are introduced and the co-circularity of a graph, G, is defined as η(G) = max{n ε N| there is a co-admissible map g:V(G)→Z<sub>n</sub>}. It is shown that σ(G) = 2η(G) or 2η(G) + 1. It is also shown that if G is a graph and g:V(G)→Z<sub>n</sub> is a co-admissible map, then G contains a cycle, J, called a co-admissible cycle, for which g:V(J)→Z<sub>n</sub> is also co-admissible. Necessary and sufficient conditions are given for extending a co-admissible map on a cycle of a graph to the entire graph. If G is a graph with σ(G) = r, it is shown that any suspended (v,w)-path P in G induces, under any admissible map f:V(G)→A(r), either at most four elements of Z<sub>r</sub> or every vertex of P with valency two induces exactly two elements of Z<sub>r</sub> not induced by any other vertex of G. Finally it is shown that if G is a planar graph and if g:V(G)→Z<sub>n</sub> is a co-admissible map, then any planar representation of G has exactly two faces bounded by co-admissible cycles."],"dc:description.degree":["Doctor of Philosophy"],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/88719"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Circularity of graphs"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:18:46Z"}