#### Defense Date

2011

#### Document Type

Thesis

#### Degree Name

Master of Science

#### Department

Mathematical Sciences

#### First Advisor

Dewey Taylor

#### Abstract

An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph’s vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We first prove that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we prove that the n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set.

#### Rights

© The Author

#### Is Part Of

VCU University Archives

#### Is Part Of

VCU Theses and Dissertations

#### Date of Submission

July 2011