#### 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.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Oellermann, Ortrud R., "Generalized Connectivity in Graphs" (1986). *Dissertations*. 2273.

http://scholarworks.wmich.edu/dissertations/2273

## Comments

Fifth Advisor: Dr. Anthony Gioia