This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.
Permalink
https://hdl.handle.net/2142/19190
Description
Title
Graph minors and algorithms
Author(s)
McGuinness, Patrick James
Issue Date
1992
Doctoral Committee Chair(s)
Brown, Donna J.
Department of Study
Mathematics
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Computer Science
Language
eng
Abstract
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.
Use this login method if you
don't
have an
@illinois.edu
email address.
(Oops, I do have one)
IDEALS migrated to a new platform on June 23, 2022. If you created
your account prior to this date, you will have to reset your password
using the forgot-password link below.
