{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/95378"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/95378","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On some problems in reconstruction","abstract":"A graph is {\\it reconstructible} if it is determined by its {\\it deck} of unlabeled subgraphs obtained by deleting one vertex; a {\\it card} is one of these subgraphs. The {\\it Reconstruction Conjecture} asserts that all graphs with at least three vertices are reconstructible. In Chapter $2$ we consider $k$-deck reconstruction of graphs. The {\\it $k$-deck} of a graph is its multiset of $k$-vertex induced subgraphs. We prove a generalization of a result by Bollob\\'as concerning the $k$-deck reconstruction of almost all graphs, showing that when $\\ell \\le (1-\\epsilon)\\frac{n}{2}$, the probability than an $n$-vertex graph is reconstructible from some $\\binom{\\ell+1}{2}$ of the graphs in the $(n-\\ell)$-deck tends to $1$ as $n$ tends to $\\infty$. We determine the smallest $k$ such that all graphs with maximum degree $2$ are $k$-deck reconstructible. We prove for $n\\ge 26$ that whether a graph is connected is determined by its $(n-3)$-deck. We prove that if $G$ is a complete $r$-partite graphs, then $G$ is $(r+1)$-deck reconstructible (the same holds for $\\overline{G}$). In Chapter $3$ we consider degree-associated reconstruction. An $(n-1)$-vertex induced subgraph accompanied with the degree of the missing vertex is called a {\\it dacard}. The {\\it degree-associated reconstruction number} of a graph $G$ is the fewest number of dacards needed to determine $G$. We provide a tool for reconstructing some graphs from two dacards. We prove that certain families of trees and disconnected graphs can be reconstructed from two dacards. We also determine the degree-associated reconstruction number for complete multipartite graphs and their complements. For such graphs, we also determine the least $s$ such that {\\it every} set of $s$ dacards determine the graph. In Chapter $4$ we consider the reconstruction of matrices from principal submatrices. A $(n-\\ell)$-by-$(n-\\ell)$ principal submatrix is a submatrix formed by deleting $\\ell$ rows and columns symmetrically. The {\\it matrix reconstruction threshold} $mrt(\\ell)$ is the minimum integer $n_0$ such that for $n\\ge n_0$ all $n$-by-$n$ matrices are reconstructible from their deck of $(n-\\ell)$-by-$(n-\\ell)$ principal submatrices. We prove $mrt(\\ell) \\leq \\frac{2}{\\ln 2}\\ell^2+3\\ell$.","abstract_html":"A graph is {\\it reconstructible} if it is determined by its {\\it deck} of unlabeled subgraphs obtained by deleting one vertex; a {\\it card} is one of these subgraphs. The {\\it Reconstruction Conjecture} asserts that all graphs with at least three vertices are reconstructible. In Chapter $2$ we consider $k$-deck reconstruction of graphs. The {\\it $k$-deck} of a graph is its multiset of $k$-vertex induced subgraphs. We prove a generalization of a result by Bollob\\&#x27;as concerning the $k$-deck reconstruction of almost all graphs, showing that when <span class=\"etd-inline-math\">\\ell \\le (1-&epsilon;)\\frac{n}{2}</span>, the probability than an $n$-vertex graph is reconstructible from some $\\binom{\\ell+1}{2}$ of the graphs in the $(n-\\ell)$-deck tends to $1$ as $n$ tends to $\\infty$. We determine the smallest $k$ such that all graphs with maximum degree $2$ are $k$-deck reconstructible. We prove for $n\\ge 26$ that whether a graph is connected is determined by its $(n-3)$-deck. We prove that if $G$ is a complete $r$-partite graphs, then $G$ is $(r+1)$-deck reconstructible (the same holds for $\\overline{G}$). In Chapter $3$ we consider degree-associated reconstruction. An $(n-1)$-vertex induced subgraph accompanied with the degree of the missing vertex is called a {\\it dacard}. The {\\it degree-associated reconstruction number} of a graph $G$ is the fewest number of dacards needed to determine $G$. We provide a tool for reconstructing some graphs from two dacards. We prove that certain families of trees and disconnected graphs can be reconstructed from two dacards. We also determine the degree-associated reconstruction number for complete multipartite graphs and their complements. For such graphs, we also determine the least $s$ such that {\\it every} set of $s$ dacards determine the graph. In Chapter $4$ we consider the reconstruction of matrices from principal submatrices. A $(n-\\ell)$-by-$(n-\\ell)$ principal submatrix is a submatrix formed by deleting $\\ell$ rows and columns symmetrically. The {\\it matrix reconstruction threshold} $mrt(\\ell)$ is the minimum integer <span class=\"etd-inline-math\">n<sub>0</sub></span> such that for <span class=\"etd-inline-math\">n\\ge n<sub>0</sub></span> all $n$-by-$n$ matrices are reconstructible from their deck of $(n-\\ell)$-by-$(n-\\ell)$ principal submatrices. We prove <span class=\"etd-inline-math\">mrt(\\ell) \\leq \\frac{2}{\\ln 2}\\ell<sup>2</sup>+3\\ell</span>.","abstract_has_math":true,"creators":["Spinoza, Hannah R"],"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","Kostochka, Alexandr","Schenck, Hal","Molla, Theo"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-03-01T15:49:22Z","date_published":"2017-03-01T15:49:22Z","updated_at":"2026-07-22T22:26:37Z","subjects":["Graph theory","Reconstruction"],"languages":["en"],"rights":["Copyright 2016 Hannah Spinoza"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/95378","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["West, Douglas","Kostochka, Alexandr","Schenck, Hal","Molla, Theo"]},{"key":"dc:creator","label":"Author","values":["Spinoza, Hannah R"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-03-01T15:49:22Z","2016-12-01","2016-12"]},{"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":["Graph theory","Reconstruction"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2016 Hannah Spinoza"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/95378"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A graph is {\\it reconstructible} if it is determined by its {\\it deck} of unlabeled subgraphs obtained by deleting one vertex; a {\\it card} is one of these subgraphs. The {\\it Reconstruction Conjecture} asserts that all graphs with at least three vertices are reconstructible. In Chapter $2$ we consider $k$-deck reconstruction of graphs. The {\\it $k$-deck} of a graph is its multiset of $k$-vertex induced subgraphs. We prove a generalization of a result by Bollob\\'as concerning the $k$-deck reconstruction of almost all graphs, showing that when $\\ell \\le (1-\\epsilon)\\frac{n}{2}$, the probability than an $n$-vertex graph is reconstructible from some $\\binom{\\ell+1}{2}$ of the graphs in the $(n-\\ell)$-deck tends to $1$ as $n$ tends to $\\infty$. We determine the smallest $k$ such that all graphs with maximum degree $2$ are $k$-deck reconstructible. We prove for $n\\ge 26$ that whether a graph is connected is determined by its $(n-3)$-deck. We prove that if $G$ is a complete $r$-partite graphs, then $G$ is $(r+1)$-deck reconstructible (the same holds for $\\overline{G}$). In Chapter $3$ we consider degree-associated reconstruction. An $(n-1)$-vertex induced subgraph accompanied with the degree of the missing vertex is called a {\\it dacard}. The {\\it degree-associated reconstruction number} of a graph $G$ is the fewest number of dacards needed to determine $G$. We provide a tool for reconstructing some graphs from two dacards. We prove that certain families of trees and disconnected graphs can be reconstructed from two dacards. We also determine the degree-associated reconstruction number for complete multipartite graphs and their complements. For such graphs, we also determine the least $s$ such that {\\it every} set of $s$ dacards determine the graph. In Chapter $4$ we consider the reconstruction of matrices from principal submatrices. A $(n-\\ell)$-by-$(n-\\ell)$ principal submatrix is a submatrix formed by deleting $\\ell$ rows and columns symmetrically. The {\\it matrix reconstruction threshold} $mrt(\\ell)$ is the minimum integer $n_0$ such that for $n\\ge n_0$ all $n$-by-$n$ matrices are reconstructible from their deck of $(n-\\ell)$-by-$(n-\\ell)$ principal submatrices. We prove $mrt(\\ell) \\leq \\frac{2}{\\ln 2}\\ell^2+3\\ell$.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-02-28 without embargo terms","The student, Hannah Spinoza, accepted the attached license on 2016-11-30 at 16:59.","The student, Hannah Spinoza, submitted this Dissertation for approval on 2016-11-30 at 17:12.","This Dissertation was approved for publication on 2016-12-01 at 14:12.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10384 on 2017-02-28 at 14:55:02","Made available in DSpace on 2017-03-01T15:49:22Z (GMT). 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In Chapter $2$ we consider $k$-deck reconstruction of graphs. The {\\it $k$-deck} of a graph is its multiset of $k$-vertex induced subgraphs. We prove a generalization of a result by Bollob\\'as concerning the $k$-deck reconstruction of almost all graphs, showing that when $\\ell \\le (1-\\epsilon)\\frac{n}{2}$, the probability than an $n$-vertex graph is reconstructible from some $\\binom{\\ell+1}{2}$ of the graphs in the $(n-\\ell)$-deck tends to $1$ as $n$ tends to $\\infty$. We determine the smallest $k$ such that all graphs with maximum degree $2$ are $k$-deck reconstructible. We prove for $n\\ge 26$ that whether a graph is connected is determined by its $(n-3)$-deck. We prove that if $G$ is a complete $r$-partite graphs, then $G$ is $(r+1)$-deck reconstructible (the same holds for $\\overline{G}$). In Chapter $3$ we consider degree-associated reconstruction. An $(n-1)$-vertex induced subgraph accompanied with the degree of the missing vertex is called a {\\it dacard}. The {\\it degree-associated reconstruction number} of a graph $G$ is the fewest number of dacards needed to determine $G$. We provide a tool for reconstructing some graphs from two dacards. We prove that certain families of trees and disconnected graphs can be reconstructed from two dacards. We also determine the degree-associated reconstruction number for complete multipartite graphs and their complements. For such graphs, we also determine the least $s$ such that {\\it every} set of $s$ dacards determine the graph. In Chapter $4$ we consider the reconstruction of matrices from principal submatrices. A $(n-\\ell)$-by-$(n-\\ell)$ principal submatrix is a submatrix formed by deleting $\\ell$ rows and columns symmetrically. The {\\it matrix reconstruction threshold} $mrt(\\ell)$ is the minimum integer $n_0$ such that for $n\\ge n_0$ all $n$-by-$n$ matrices are reconstructible from their deck of $(n-\\ell)$-by-$(n-\\ell)$ principal submatrices. We prove $mrt(\\ell) \\leq \\frac{2}{\\ln 2}\\ell^2+3\\ell$.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-02-28 without embargo terms","The student, Hannah Spinoza, accepted the attached license on 2016-11-30 at 16:59.","The student, Hannah Spinoza, submitted this Dissertation for approval on 2016-11-30 at 17:12.","This Dissertation was approved for publication on 2016-12-01 at 14:12.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10384 on 2017-02-28 at 14:55:02","Made available in DSpace on 2017-03-01T15:49:22Z (GMT). No. of bitstreams: 3 SPINOZA-DISSERTATION-2016.pdf: 711995 bytes, checksum: 7e5c88498aec939a8a2df32190192036 (MD5) LICENSE.txt: 4211 bytes, checksum: db6cd84f8fde9b51cdd073ef0b3a0a9a (MD5) PROQUEST_LICENSE.txt: 4557 bytes, checksum: b1225763f7a2802f350edb05f454583f (MD5) Previous issue date: 2016-12-01"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/95378"],"dc:language":["en"],"dc:rights":["Copyright 2016 Hannah Spinoza"],"dc:subject":["Graph theory","Reconstruction"],"dc:title":["On some problems in reconstruction"],"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":"2026-07-22T22:26:37Z"}