Added Graph Algorithms handout

Advanced/Graph Algorithms/parts/00 review.tex

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+\documentclass[../main.tex]{subfiles}
+
+
+\begin{document}
+
+	\section{Review}
+
+	\definition{}
+	A \textit{graph} consists of a set of \textit{nodes} $\{A, B, ...\}$ and a set of edges $\{ (A,B), (A,C), ...\}$ connecting them.
+	A \textit{directed graph} is a graph where edges have direction. In such a graph, $(A, B)$ and $(B, A)$ are distinct edges.
+	A \textit{weighted graph} is a graph that features weights on its edges. \\
+	A weighted directed graph is shown below.
+
+	\begin{center}
+		\begin{tikzpicture}[node distance = 20mm]
+			% Nodes
+			\begin{scope}
+				\node[main] (A) {$A$};
+				\node[main] (B) [below right of = A] {$B$};
+				\node[main] (C) [below left of = A] {$C$};
+			\end{scope}
+
+			% Edges
+			\draw[->]
+				(A) edge[bend right] node[label] {$4$} (B)
+				(B) edge node[label] {$2$} (C)
+				(C) edge node[label] {$2$} (A)
+				(B) edge[bend right] node[label] {$1$} (A)
+			;
+		\end{tikzpicture}
+	\end{center}
+
+	\vfill
+
+	\definition{}
+	We say a graph is \textit{bipartite} if its nodes can be split into two groups $L$ and $R$ so that no two nodes in the same group are connected by an edge. \\
+	The following graph is bipartite:
+
+	\begin{center}
+		\begin{tikzpicture}
+			% Nodes
+			\begin{scope}
+				\node[main] (A) at (0mm, 0mm) {$A$};
+				\node[main] (B) at (0mm, -10mm) {$B$};
+				\node[main] (C) at (0mm, -20mm) {$C$};
+
+				\node[main] (D) at (20mm, 0mm) {$D$};
+				\node[main] (E) at (20mm, -10mm) {$E$};
+				\node[main] (F) at (20mm, -20mm) {$F$};
+			\end{scope}
+
+			% Edges
+			\draw
+				(A) edge (D)
+				(A) edge (E)
+				(B) edge (F)
+				(C) edge (E)
+				(C) edge (D)
+			;
+		\end{tikzpicture}
+	\end{center}
+
+	\vfill
+
+	\definition{}
+	We say two nodes $A$ ane $B$ are \textit{connected} if we can reach $A$ from $B$ and $B$ from $A$ by walking along (possibly directed) edges. We say a graph is connected if all its nodes are connected to each other.\\
+
+	The bipartite graph above and the directed graph below are not connected.
+
+	\begin{center}
+		\begin{tikzpicture}[node distance = 20mm]
+			% Nodes
+			\begin{scope}
+				\node[main] (A) {$A$};
+				\node[main] (B) [below right of = A] {$B$};
+				\node[main] (C) [below left of = A] {$C$};
+			\end{scope}
+
+			% Edges
+			\draw[->]
+				(A) edge[bend right] (B)
+				(B) edge[bend right] (A)
+				(B) edge (C)
+			;
+		\end{tikzpicture}
+	\end{center}
+
+	\vfill
+	\pagebreak
+
+\end{document}