{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71264"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71264","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Total Interval Number of a Graph","abstract":"An interval representation (or simply representation) R of a graph G is a collection of finite sets $\\{R(\\nu):\\nu \\in V(G)\\}$ of closed bounded intervals so that $u \\leftrightarrow \\nu$ if and only if there exist $\\theta\\sb{u} \\in R(u), \\theta\\sb{\\nu} \\in R(\\nu)$ with $\\theta\\sb{u} \\cap \\theta\\sb{\\nu} \\not= \\emptyset$. The size of a representation is the number of intervals in the entire collection.","abstract_html":"An interval representation (or simply representation) R of a graph G is a collection of finite sets $\\{R(\\nu):\\nu \\in V(G)\\}$ of closed bounded intervals so that $u \\leftrightarrow \\nu$ if and only if there exist <span class=\"etd-inline-math\">&theta;\\sb{u} \\in R(u), &theta;\\sb{\\nu} \\in R(\\nu)</span> with <span class=\"etd-inline-math\">&theta;\\sb{u} \\cap &theta;\\sb{\\nu} \\not= \\emptyset</span>. The size of a representation is the number of intervals in the entire collection.","abstract_has_math":true,"creators":["Kratzke, Thomas Martin"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["West, Douglas B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:21Z","date_published":"2014-12-16T06:18:21Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics","Computer Science"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8815372"],"render_values":[{"text":"(UMI)AAI8815372","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71264","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["West, Douglas B."]},{"key":"dc:creator","label":"Author","values":["Kratzke, Thomas Martin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:21Z","10000-01-01","1988"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Computer Science"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71264","(UMI)AAI8815372"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["An interval representation (or simply representation) R of a graph G is a collection of finite sets $\\{R(\\nu):\\nu \\in V(G)\\}$ of closed bounded intervals so that $u \\leftrightarrow \\nu$ if and only if there exist $\\theta\\sb{u} \\in R(u), \\theta\\sb{\\nu} \\in R(\\nu)$ with $\\theta\\sb{u} \\cap \\theta\\sb{\\nu} \\not= \\emptyset$. The size of a representation is the number of intervals in the entire collection.","The total interval number of G is the size of the smallest representation of G and is denoted I(G). This thesis studies I by proving best possible upper bounds for several classes of graphs. For some classes, the bounds are in terms of n, the number of vertices and for some classes, the bounds are in terms of m, the number of edges. The main result is that for planar graphs, $I(G) \\leq 2n(G) - 3$.","Made available in DSpace on 2014-12-16T06:18:21Z (GMT). No. of bitstreams: 1 8815372.pdf: 5483196 bytes, checksum: 87e060f0b586a5ddab657ced37f050da (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71430 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","123 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."]},{"key":"dc:title","label":"Title","values":["The Total Interval Number of a Graph"]}]}],"canonical_facts":{"dc:contributor":["West, Douglas B."],"dc:creator":["Kratzke, Thomas Martin"],"dc:date":["2014-12-16T06:18:21Z","10000-01-01","1988"],"dc:description":["An interval representation (or simply representation) R of a graph G is a collection of finite sets $\\{R(\\nu):\\nu \\in V(G)\\}$ of closed bounded intervals so that $u \\leftrightarrow \\nu$ if and only if there exist $\\theta\\sb{u} \\in R(u), \\theta\\sb{\\nu} \\in R(\\nu)$ with $\\theta\\sb{u} \\cap \\theta\\sb{\\nu} \\not= \\emptyset$. The size of a representation is the number of intervals in the entire collection.","The total interval number of G is the size of the smallest representation of G and is denoted I(G). This thesis studies I by proving best possible upper bounds for several classes of graphs. For some classes, the bounds are in terms of n, the number of vertices and for some classes, the bounds are in terms of m, the number of edges. The main result is that for planar graphs, $I(G) \\leq 2n(G) - 3$.","Made available in DSpace on 2014-12-16T06:18:21Z (GMT). No. of bitstreams: 1 8815372.pdf: 5483196 bytes, checksum: 87e060f0b586a5ddab657ced37f050da (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71430 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","123 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."],"dc:identifier":["http://hdl.handle.net/2142/71264","(UMI)AAI8815372"],"dc:subject":["Mathematics","Computer Science"],"dc:title":["The Total Interval Number of a Graph"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}