-Directed graphs are special kinds of graphs. A directed graph is described as
-the ordered pair: $G=(V,A)$. Where $A$ is a set of ordered pairs that we will
-call \textit{directed edges}. Directed edges can be visualized as arrows in a
-graph whereas undirected edges are just lines connecting the nodes with no
-particular direction.
-
-Directed Acyclic Graphs(DAGs) are again a special kind of directed graph. DAGs
-don't allow cycles to be created. Every node is only reachable at maximum once
-when starting from an arbitrary point in the graph. DAGs have the nice property
-of checking if a sequence appears in the graph, checking if a sequence is
-present in a graph only has a computational complexity of $\mathcal{O}(L)$
-where $L$ is the length of the sequence.
-
-The type of graph used in the project is another special king of DAG called
-Directed Acyclic Word Graphs(DAWGs). This type of graph can be used to store
-large dictionaries of words. A DAWGs nodes basically represent indices of a
-word and the edges describe the character added when traversing. For example
-the graph in Figure~\ref{fig:2.1.1} can describe a dictionary that holds the
-words: \textit{abd, bad, bae}. Testing if a word is present in the DAWG is,
-just as for a DAG, falls als in the computational complexity class of
-$\mathcal{O}(L)$ meaning that it grows linearly with the length of the word.
+\begin{figure}[H]
+ \caption{Example DAG}
+ \label{fig:dagexample}
+ \centering
+ \digraph[]{dagexample}{
+ rankdir=LR
+ n01 -> n02
+ n02 -> n03
+ n03 -> n01
+ n11 -> n12
+ n12 -> n13
+ n12 -> n14
+ }
+\end{figure}
+
+\paragraph{Directed Acyclic Word Graphs}
+The type of graph used in the project is a special kind of DAG called
+Directed Acyclic Word Graphs(DAWGs). A DAWG can be defined by the ordered pair
+$G=(V,E)$ and is the same as a directed graph except for the edges. An edge in
+a DAWG is instead of a tuple a triple and consist of a starting point, an end
+point and a label. Because of the property of labeled edges data can be stored
+in a DAWG. When traversing a DAWG and saving all the edgelabels one can
+construct words. Because of properties of graphs a DAWG can store big sets of
+words in a small amount of storage because it can re-use edges to specify
+transitions. For example the graph in Figure~\ref{fig:2.1.1} can describe a
+dictionary that holds the words: \textit{abd, bad, bae}. Testing if a word is
+present in the DAWG is, just as for a DAG, falls in the computational
+complexity class of $\mathcal{O}(L)$ meaning that it grows linearly with the
+length of the word.