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oai:www.ideals.illinois.edu:2142/19190 2023-07-10 col_2142_5131 col_2142_10761 com_2142_5130 com_2142_10755 com_2142_234

Brown, Donna J. McGuinness, Patrick James 2011-05-07T11:59:41Z 2011-05-07T11:59:41Z 10000-01-01 1992 A graph H is a minor of another graph G, denoted by $H\ {\prec\sb{m}}\ G,$ if a graph isomorphic to H can be obtained from G by a series of vertex deletions, edge deletions, and edge contractions. Graph minors have been studied for several decades as a way of characterizing classes of graphs. Recent work by Robertson and Seymour has provided further motivation for studying both mathematical and computational aspects of graph minor theory. This thesis examines some graph-theoretic and algorithmic aspects of graph minors. We examine connectivity, minimum degree, and related minor-ordered functions. In particular, we study the sets of minor-minimal graphs for minimum degree and connectivity 4, 5, and 6, and present several classes of graphs that are minor-minimal for connectivity and minimum degree k, for general values of k. We present sequential and parallel algorithms to test for a $K\sb5$ minor. In the process of describing these algorithms, we prove structural results concerning graphs that do not contain a $K\sb5$ minor. Our $O(n\sp2)$ sequential algorithm tests for the existence of a $K\sb5$ minor in a graph and, if a $K\sb5$ minor exists, returns the branch sets of a $K\sb5$ minor. Our parallel algorithm to find a $K\sb5$ minor in a graph requires $O(\log\sp2 n)$ time and $O(n\sp3\alpha(n, n)/\log\ n)$ processors. Following up on our $K\sb5$ minor algorithm, we examine classes of graphs that do not contain a $K\sb6$ minor. Finally, we examine pathwidth, a minor-ordered function that plays an important role in the work of Robertson and Seymour. We show bounds relating pathwidth, cutwidth and treewidth, and we present a characterization of graphs with pathwidth k, for any value of k. Made available in DSpace on 2011-05-07T11:59:41Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9236539.pdf: 5436075 bytes, checksum: 5d09e62d283cd63f056eba6b089fac4c (MD5) Previous issue date: 1992 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:35:16Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:13:50-05:00 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 AAI9236539 (UMI)AAI9236539 http://hdl.handle.net/2142/19190 eng Copyright 1992 McGuinness, Patrick James Mathematics Computer Science Graph minors and algorithms text Mathematics Computer Science Computer Science University of Illinois at Urbana-Champaign Dissertation Ph.D.