{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21303"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21303","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Graph representations using stars, trees, intervals and boxes","abstract":"We introduce star number (tree number) of a graph G, which is the minimum t such that G is the intersection graph of unions of t substars (subtrees) of a host tree. We characterize the graphs with star number 1 and prove that a planar graph has star number at most 3. We study bounds on these two parameters and compare them with interval number. We prove that the star number is at most $\\lceil(n + 1)/4\\rceil,$ where n is the number of vertices. We also show the independence of interval number and star number.","abstract_html":"We introduce star number (tree number) of a graph G, which is the minimum t such that G is the intersection graph of unions of t substars (subtrees) of a host tree. We characterize the graphs with star number 1 and prove that a planar graph has star number at most 3. We study bounds on these two parameters and compare them with interval number. We prove that the star number is at most $\\lceil(n + 1)/4\\rceil,$ where n is the number of vertices. We also show the independence of interval number and star number.","abstract_has_math":true,"creators":["Chang, Yi-Wu"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Weischel, Paul W."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:04:42Z","date_published":"2011-05-07T13:04:42Z","updated_at":"2026-07-22T22:25:17Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1994 Chang, Yi-Wu"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9512323","(UMI)AAI9512323"],"render_values":[{"text":"AAI9512323","href":null,"code":true},{"text":"(UMI)AAI9512323","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21303","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Weischel, Paul W."]},{"key":"dc:creator","label":"Author","values":["Chang, Yi-Wu"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:04:42Z","10000-01-01","1994"]},{"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"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1994 Chang, Yi-Wu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9512323","(UMI)AAI9512323","http://hdl.handle.net/2142/21303"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We introduce star number (tree number) of a graph G, which is the minimum t such that G is the intersection graph of unions of t substars (subtrees) of a host tree. We characterize the graphs with star number 1 and prove that a planar graph has star number at most 3. We study bounds on these two parameters and compare them with interval number. We prove that the star number is at most $\\lceil(n + 1)/4\\rceil,$ where n is the number of vertices. We also show the independence of interval number and star number.","We also prove some results about representations using intervals and higher-dimensional objects. The rectangle number of G is the minimum t such that G has an intersection representation in which each vertex is assigned a union of t boxes in the plane. The rectangle number of a multipartite graph is at most 2. The rectangle number of a k-dimensional cube is at most $\\lceil k/4\\rceil,$ except for k = 4. For an intersection representation of a digraph, we assign a source set $S\\sb u$ and a sink set $T\\sb u$ to each vertex u such that uv is an edge if and only if $S\\sb u\\cap T\\sb v\\ne\\emptyset.$ The interval number of a digraph is the minimum t such that D has an intersection representation in which each source set and sink set is a union of t intervals. We prove that the interval number of a digraph is at most n/(lgn + 1). The bar visibility number of a graph G is the minimum t such that G has an representation in which each vertex is assigned a union of t horizontal intervals (bars) in the plane such that two vertices u,v are adjacent if and only if some bar for u can see some bar for v by an unblocked vertical line. We prove that the bar visibility number is at most $\\lceil n/6\\rceil + 2$ for graphs with n vertices.","Made available in DSpace on 2011-05-07T13:04:42Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512323.pdf: 3860089 bytes, checksum: 237a2ede4d7f0eb994e49dc3e8992142 (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:49:51Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:22:43-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Graph representations using stars, trees, intervals and boxes"]}]}],"canonical_facts":{"dc:contributor":["Weischel, Paul W."],"dc:creator":["Chang, Yi-Wu"],"dc:date":["2011-05-07T13:04:42Z","10000-01-01","1994"],"dc:description":["We introduce star number (tree number) of a graph G, which is the minimum t such that G is the intersection graph of unions of t substars (subtrees) of a host tree. We characterize the graphs with star number 1 and prove that a planar graph has star number at most 3. We study bounds on these two parameters and compare them with interval number. We prove that the star number is at most $\\lceil(n + 1)/4\\rceil,$ where n is the number of vertices. We also show the independence of interval number and star number.","We also prove some results about representations using intervals and higher-dimensional objects. The rectangle number of G is the minimum t such that G has an intersection representation in which each vertex is assigned a union of t boxes in the plane. The rectangle number of a multipartite graph is at most 2. The rectangle number of a k-dimensional cube is at most $\\lceil k/4\\rceil,$ except for k = 4. For an intersection representation of a digraph, we assign a source set $S\\sb u$ and a sink set $T\\sb u$ to each vertex u such that uv is an edge if and only if $S\\sb u\\cap T\\sb v\\ne\\emptyset.$ The interval number of a digraph is the minimum t such that D has an intersection representation in which each source set and sink set is a union of t intervals. We prove that the interval number of a digraph is at most n/(lgn + 1). The bar visibility number of a graph G is the minimum t such that G has an representation in which each vertex is assigned a union of t horizontal intervals (bars) in the plane such that two vertices u,v are adjacent if and only if some bar for u can see some bar for v by an unblocked vertical line. We prove that the bar visibility number is at most $\\lceil n/6\\rceil + 2$ for graphs with n vertices.","Made available in DSpace on 2011-05-07T13:04:42Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512323.pdf: 3860089 bytes, checksum: 237a2ede4d7f0eb994e49dc3e8992142 (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:49:51Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:22:43-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9512323","(UMI)AAI9512323","http://hdl.handle.net/2142/21303"],"dc:language":["eng"],"dc:rights":["Copyright 1994 Chang, Yi-Wu"],"dc:subject":["Mathematics"],"dc:title":["Graph representations using stars, trees, intervals and boxes"],"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":"2026-07-22T22:25:17Z"}