% % % % % % %
% %Abstract % %
% % % % % % %

\begin{center}
Abstract 

\bigskip

\myTitle \\
By \myName

\end{center}

\bigskip

%There is a 350 word limit.  This page can be single spaced if needed.

Given a graph $H$, we say a graph $G$ is {\it H-saturated} if $G$ does not contain $H$ as a subgraph and the addition of any edge $e\not\in E(G)$ results in $H$ as a subgraph. The question of the minimum number of edges of an $H$-saturated graph on $n$ vertices, known as the {\it saturation number}, and the question of the maximum number of edges possible of an $H$-saturated graph, known as the Tur\'an number, have been addressed for many different types of graphs. We are interested in the existence of $H$-saturated graphs for each edge count between the saturation number and the Tur\'an number. 

In this thesis, we determine the saturation spectrum of $(K_t-e)$-saturated graphs and $F_t$-saturated graphs. Let $K_t-e$ be the complete graph minus one edge. We prove that $(K_4-e)$-saturated graphs do not exist for small values of $|E(G)|$ and construct $(K_4-e)$-saturated graphs with $|E(G)|$ in the interval $\left[2n-4, \left\lfloor \frac{n}{2}\right\rfloor\left\lceil\frac{n}{2}\right\rceil-\left\lfloor\frac{n}{2}\right\rfloor+1\right]$. We then extend the constructed $(K_4-e)$-saturated graphs to create $(K_t-e)$-saturated graphs. 

Let $F_t$ be the graph consisting of $t$ edge-disjoint triangles that intersect at a single vertex $v$. We prove that $F_2$-saturated graphs do not exist for small edge counts and construct a collection of $F_2$-saturated graphs with edge counts in the interval $[2n-4, \frac{n^2}{4}-\lfloor\frac{n}{2}\rfloor+2]$ or the size of a complete bipartite graph with an additional edge. We also establish two general constructions that yield $F_t$-saturated graphs with edge counts in $\left[\left(\frac{3t}{2}-1\right)n-\left\lfloor\frac{3t}{2}\right\rfloor\left\lceil\frac{3t}{2}\right\rceil+t^2-4, \left\lceil\frac{n}{2}\right\rceil\left\lfloor\frac{n}{2}\right\rfloor-\left\lfloor\frac{n}{2}\right\rfloor+2+(t-1)^2\right].$
\newpage
