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 vertexu belongs to a copy of F rooted at u . The F -domination number γF (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 γF (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 γF (G ) = a and γr (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 γH and both the domination number γ and open domination γ o in digraphs. (Abstract shortened by UMI.)

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Dissertation-Open Access

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