Date of Award

8-1986

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dr. Gary Chartrand

Second Advisor

Dr. Shashi Kapoor

Third Advisor

Dr. Yousef Alavi

Fourth Advisor

Dr. Raymond Pippert

Abstract

The connectivity of a graph G is the minimum number of vertices in G whose deletion produces a disconnected or trivial graph, while the edge-connectivity of G is the minimum number of edges having this property. In this dissertation several generalizations and variations of these two parameters are introduced and studied.

Chapter I is an overview to the history of connectivity and provides a background for the chapters that follow. In Chapter II major n-connected subgraphs are introduced. Through this concept, the connectivities (of subgraphs) that are most representative in a given graph are studied.

Chapter III is devoted to the study of determining the minimum cardinality of a vertex cutset S of a graph G, with given induced sub- graph F, such that either each component of G - S is a subgraph of F or each component is F-free. The corresponding edge versions of these concepts are also introduced and studied. Among the results presented are analogues of the well-known connectivity theorem of Whitney.

Two variations of connectivity are studied in Chapter IV. First, the L-connectivity (L (GREATERTHEQ) 2) of a graph G is considered, which is the minimum number of vertices which need to be deleted from G to produce a disconnected graph with at least L components or a graph with fewer than L vertices. A graph is (n,L)-connected if its L-connectivity is at least n. Several sufficient conditions for a graph to be (n,L)-connected are established. Second, the connected cutset connectivity of a graph G is introduced and defined as the minimum cardinality of a vertex cutset S of G such that S induces a connected subgraph. The connected edge-cutset connectivity is defined simi-larly. The connected edge-cutset connectivity is bounded below by the edge-connectivity and above by the minimum degree of a graph. As an analogue to a well-known problem in connectivity, a sufficient condition is presented for which the minimum degree equals the connected edge-cutset connectivity but does not necessarily equal the edge-connectivity.

Comments

Fifth Advisor: Dr. Anthony Gioia

Access Setting

Dissertation-Open Access

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