#### Date of Award

4-2005

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### Abstract

In this thesis we combine the idea of stratification with the one of domination in graphs and digraphs, respectively.

A graph is 2-stratified if its vertex set is partitioned into two classes, where the vertices in one class are colored red and those in the other class are colored blue. Let *F* be a 2-stratified graph rooted at some blue vertex *v* . An *F* -coloring of a graph *G* is a red-blue coloring of the vertices of *G* in which every blue vertex*u* belongs to a copy of *F *rooted at *u* . The *F* -domination number γ_{F} (*G* ) is the minimum number of red vertices in an *F* -coloring of *G* .

It is shown in Chapter 3 that (1) for each pair *a, b* of positive integers, there exists a connected graph *G* such that γ(*G* ) = *a* and γ* _{F}* (

*G*) =

*b*; (2) for each pair

*a, b*of positive integers with

*a*≥ 2, there exists a connected graph

*G*such that γ

*o*(

*G*) =

*a*and γ

*F*(

*G*) =

*b*; (3) for each pair

*a, b*of positive integers with

*a*≤

*b*, there exists a connected graph

*G*such that γ ∃2 (

*G*) =

*a*and γ

*(*

_{F}*G*) =

*b*if and only if (

*a, b*) ≠ (1,

*i*) for some

*i*≥ 2; and (4) for each pair

*a, b*of positive integers with

*a*≤

*b*, there exists a connected graph

*G*with γ

*F*(

*G*) =

*a*and γ

*r*(

*G*) =

*b*. In Chapter 4, we show that a triple ([Special characters omitted.] ) of positive integers with [Special characters omitted.] and [Special characters omitted.] ≥ 2 is realizable as the domination number, open domination number, and

*F*-dominationnumber, respectively, for some connected graph if and only if ([Special characters omitted.] ) ≠ (

*k, k*, [Special characters omitted.] ) for integers

*k*and [Special characters omitted.] with [Special characters omitted.] >

*k*≥ 2.

In Chapter 5, *H* -domination is studied where *H* is the red-red-blue directed path of order 3. We study relationships between the *H* -domination number γ*H* and both the domination number γ and open domination γ* o* in digraphs. (Abstract shortened by UMI.)

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Gera, Ralucca M., "Stratification and Domination in Graphs and Digraphs" (2005). *Dissertations*. 1033.

http://scholarworks.wmich.edu/dissertations/1033