Graph minors and algorithms
Patrick J. McGuinness · 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.