101 lines
3.0 KiB
TeX
101 lines
3.0 KiB
TeX
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\section{Line Graphs}
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\problem{}
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Given a graph $G$, we can construct its \textit{line graph} (denoted $\mathcal{L}(G)$) by doing the following: \par
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\begin{itemize}
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\item Creating a node in $\mathcal{L}(G)$ for each edge in $G$
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\item Drawing a directed edge between every pair of nodes $a, b$ in $\mathcal{L}(G)$ \par
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if the corresponding edges in $G$ are adjacent. \par
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\note{That is, if edge $b$ in $G$ starts at the node at which $a$ ends.}
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\end{itemize}
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\problem{}
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Draw the line graph for the graph below. \par
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Have an instructor check your solution.
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\begin{center}
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\begin{tikzpicture}
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\begin{scope}[layer = nodes]
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\node[main] (a) at (0, 0) {$a$};
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\node[main] (b) at (2, 0) {$b$};
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\node[main] (c) at (4, 0) {$c$};
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\end{scope}
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\draw[->]
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(a) edge[bend left] node[label] {$0$} (b)
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(b) edge[bend left] node[label] {$1$} (a)
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(b) edge[bend left] node[label] {$2$} (c)
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(c) edge[bend left] node[label] {$3$} (b)
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(c) edge[loop right] node[label] {$4$} (c)
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;
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\end{tikzpicture}
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\end{center}
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\begin{solution}
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\begin{center}
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\begin{tikzpicture}
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\begin{scope}[layer = nodes]
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\node[main] (0) at (0, 0) {$0$};
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\node[main] (1) at (2, -4) {$1$};
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\node[main] (2) at (0, -2) {$2$};
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\node[main] (3) at (2, -2) {$3$};
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\node[main] (4) at (2, 0) {$4$};
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\end{scope}
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\draw[->]
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(0) edge[bend left] (2)
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(2) edge[bend left] (0)
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(0) edge (4)
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(4) edge[bend left] (2)
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(2) edge (1)
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(1) edge[bend left] (3)
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(3) edge[bend left] (1)
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(3) edge (0)
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;
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\end{tikzpicture}
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\end{center}
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\end{solution}
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\vfill
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\definition{}
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We say a graph $G$ is \textit{connected} if there is a path
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between any two vertices of $G$.
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\problem{}
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Show that if $G$ is connected, $\mathcal{L}(G)$ is connected.
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\begin{solution}
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Let $a, b$ and $x, y$ be nodes in a connected graph $G$ so that an edges $a \rightarrow b$ and
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and $x \rightarrow y$ exist. Since $G$ is connected, we can find a path from $b$ to $x$.
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The path $a$ to $y$ corresponds to a path in $\mathcal{L}(G)$ between $a \rightarrow b$ and $x \rightarrow y$.
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\end{solution}
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\vfill
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\pagebreak
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\definition{}
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Consider $\mathcal{L}(G_n)$, where $G_n$ is the $n^\text{th}$ order De Bruijn graph. \par
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\vspace{2mm}
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We'll need to label the vertices of $\mathcal{L}(G_n)$. To do this, do the following:
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\begin{itemize}
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\item Let $a$ and $b$ be nodes in $G_n$
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\item Let \texttt{x} be the first letter of $a$
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\item Let \texttt{y}, the last letter of $b$
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\item Let $\overline{\texttt{p}}$ be the prefix/suffix that $a$ and $b$ share. \par
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Note that $a = \texttt{x}\overline{\texttt{p}}$ and $b = \overline{\texttt{p}}\texttt{y}$,
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\end{itemize}
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Now, relabel the edge from $a$ to $b$ as $\texttt{x}\overline{\texttt{p}}\texttt{y}$. \par
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Use these new labels to name nodes in $\mathcal{L}(G_n)$.
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\problem{}
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Construct $\mathcal{L}(G_2)$ and $\mathcal{L}(G_3)$. What do you notice?
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\begin{solution}
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After fixing edge labels, we find that
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$\mathcal{L}(G_2) \cong G_3$, and $\mathcal{L}(G_3) \cong G_4$
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\end{solution}
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