#### Title

#### Date of Award

12-1985

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Gary Chartrand

#### Second Advisor

Dr. Yousef Alavi

#### Third Advisor

Dr. Ronald J. Gould

#### Fourth Advisor

Dr. Philip Hsieh

#### Abstract

A greatest common (induced) subgraph of graphs G(,1) and G(,2) of equal size is a (an induced) common subgraph (without isolated vertices) of G(,1) and G(,2) having maximum size. For a given graph L, a graph G is locally L if the induced subgraph of the neighborhood of each vertex of G is isomorphic to L (L is called a common link). Several topics concerning these concepts are investigated in this dissertation. An historical background to these topics is given in Chapter I.

Chapter II is devoted to the topic on highly connected unique greatest common subgraphs of graphs. We show that if G is 2-connected, G (NOT=) K(,p) (p (GREATERTHEQ) 3) and G (NOT=) K(,p) - e (p (GREATERTHEQ) 4), then G is a unique greatest common subgraph of two suitably chosen graphs of equal size.

In Chapter III we determine all trees (UPSILON) for which (UPSILON) is a unique greatest common subgraph of two trees of equal size. We define the gcs tree number of a nontrivial tree and gcs number of a graph without isolated vertices. We determine these numbers for some specific graphs.

In Chapter IV we introduce the more general concepts "maximal common subgraphs" and "Least maximal common subgraphs". We describe the behavior of the least maximal common subgraph. Then we develop a lower bound for a least maximal common subgraph and show that this lower bound is sharp.

Chapter V is devoted to the common link. We develop some necessary and sufficient conditions for a graph to be a common link. Then we define the index of a common link. We find the index of L if L is the finite union of disjoint complete graphs. Furthermore, we determine the indices for many specific graphs. Finally, we briefly discuss the link number of a graph which is not a common link.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Zou, Hung Bin, "On Common Subgraphs" (1985). *Dissertations*. 2389.

http://scholarworks.wmich.edu/dissertations/2389

## Comments

Fifth Advisor: Dr. Mark Johnson