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+\begin{frame}{} + \begin{definition}[Irreducible Tree] + An \red{irreducible tree} is a tree $T$ where + \[ + \forall v \in V(T).\; \degree(v) \neq 2. + \] + \end{definition} + + \pause + \vspace{0.30cm} + \begin{columns} + \column{0.50\textwidth} + \fig{width = 0.80\textwidth}{figs/trees-10} + \column{0.50\textwidth} + \pause + \fig{width = 0.80\textwidth}{figs/trees-movie} + \end{columns} + + \pause + \vspace{0.30cm} + \begin{center} + \blue{Homeomorphically} Irreducible Trees of size $n = 10$ + \end{center} +\end{frame} %%%%%%%%%%%%%%% \ No newline at end of file diff --git a/10-trees/parts/def.tex b/10-trees/parts/def.tex index 5fdf42b..1865a15 100644 --- a/10-trees/parts/def.tex +++ b/10-trees/parts/def.tex @@ -7,7 +7,7 @@ \end{definition} \pause - \vspace{0.60cm} + \vspace{0.80cm} \begin{definition}[Forest (森林)] A \red{forest} is a \purple{acyclic} \blue{undirected} graph. \end{definition} @@ -24,14 +24,20 @@ \end{columns} \pause - \fig{width = 0.40\textwidth}{figs/Cayley-Graph} - - \pause - \vspace{0.30cm} + \vspace{0.80cm} \fig{width = 0.70\textwidth}{figs/trees} \end{frame} %%%%%%%%%%%%%%% +%%%%%%%%%%%%%%% +\begin{frame}{} + \fig{width = 0.40\textwidth}{figs/Cayley-Graph} + \begin{center} + \teal{Cayley Graph} (4-regular tree) + \end{center} +\end{frame} +%%%%%%%%%%%%%%% + %%%%%%%%%%%%%%% \begin{frame}{} \begin{definition}[Internal Vertex (内部顶点); Leaf (叶子)] @@ -99,82 +105,247 @@ \pause \vspace{0.50cm} \blue{A leaf does \red{\it not} belong to any paths connecting two other vertices.} - \end{center} -\end{frame} -%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%% -\begin{frame}{} - \begin{definition}[Irreducible Tree] - An \red{irreducible tree} is a tree $T$ where - \[ - \forall v \in V(T).\; \degree(v) \neq 2. - \] - \end{definition} - - \pause - \vspace{0.30cm} - \begin{columns} - \column{0.50\textwidth} - \fig{width = 0.80\textwidth}{figs/trees-10} - \column{0.50\textwidth} - \pause - \fig{width = 0.80\textwidth}{figs/trees-movie} - \end{columns} - - \pause - \vspace{0.30cm} - \begin{center} - \blue{Homeomorphically} Irreducible Trees of size $n = 10$ + \pause + \vspace{0.80cm} + \red{This lemma can be used in induction for trees!} \end{center} \end{frame} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \begin{frame}{} - \begin{theorem}[\cyan{(We call it)} Tree Theorem] + \begin{theorem}[\cyan{(We call it)} Characterization of Trees] Let $T$ be an undirected graph with $n$ vertices. \\[3pt] Then the following statements are \red{equivalent}: \begin{enumerate}[(1)] \setlength{\itemsep}{6pt} \item $T$ is a tree; - \item $T$ is acyclic, and has $n-1$ edges; - \item $T$ is connected, and has $n-1$ edges; + \item $T$ is acyclic, and has $m = n-1$ edges; + \item $T$ is connected, and has $m = n-1$ edges; \item $T$ is connected, and each edge is a \cyan{bridge}; \item Any two vertices of $T$ are connected by exactly one path; \item $T$ is acyclic, but the addition of any edge creates exactly one cycle. \end{enumerate} \end{theorem} + + \pause + \[ + \red{(1)} \implies (2) \implies (3) \implies (4) \implies (5) \implies (6) \implies \red{(1)} + \] \end{frame} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \begin{frame}{} - \fig{width = 0.30\textwidth}{figs/keep-calm-prove-it} + \begin{theorem}[Characterization of Trees] + \begin{enumerate}[(1)] + \setlength{\itemsep}{6pt} + \item $T$ is a tree; + \item $T$ is acyclic, and has $m = n-1$ edges. + \end{enumerate} + \end{theorem} + + \pause + \begin{center} + \red{By induction on the number $n$ of vertices of trees.} + \end{center} + + \begin{description}[Induction Hypothesis:] + \setlength{\itemsep}{8pt} + \item[Basis Step:] + \uncover<3->{$n = 1$. $m = 0 = n - 1$.} + \item[Induction Hypothesis:] + \uncover<4->{Any trees with $n-1$ vertices has $n-2$ edges.} + \item[Induction Step:] + \uncover<5->{Consider a tree $T$ with $n \ge 2$ vertices. \\[5pt]} + \uncover<6->{\blue{$T$ has a leaf $v$.} \\[5pt]} + \uncover<7->{For \red{$T' = T - v$}, $m(T') = (n-1)-1 = n-2$.} + \uncover<8->{ + \[ + m(T) = (n-2) + 1 = n - 1. + \] + } + \end{description} \end{frame} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \begin{frame}{} + \begin{theorem}[Characterization of Trees] + \begin{enumerate}[(1)] + \setcounter{enumi}{1} + \setlength{\itemsep}{6pt} + \item $T$ is acyclic, and has $n-1$ edges; + \item $T$ is connected, and has $n-1$ edges. + \end{enumerate} + \end{theorem} + + \pause + \vspace{0.30cm} + \begin{center} + \red{By Contradiction.} + + \pause + \vspace{0.30cm} + Suppose that $T$ is {\it disconnected}. + + \pause + \vspace{0.30cm} + \blue{$T$ is a forest, consisting of $k \ge 2$ trees $T_{1}, T_{2}, \dots$.} + + \pause + \vspace{0.30cm} + \red{By (2), for each $T_{i}$, $m(T_{i}) = n(T_{i}) - 1$.} + + \pause + \[ + m(T) = \sum_{i=1}^{k} m(T_{i}) = n - k \neq n - 1. + \] + \end{center} \end{frame} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \begin{frame}{} + \begin{theorem}[Characterization of Trees] + \begin{enumerate}[(1)] + \setcounter{enumi}{2} + \setlength{\itemsep}{6pt} + \item $T$ is connected, and has $n-1$ edges; + \item $T$ is connected, and each edge is a \cyan{bridge}. + \end{enumerate} + \end{theorem} + + \pause + \fig{width = 0.30\textwidth}{figs/bridge} + + \begin{definition}[Bridge (桥)] + A \red{bridge} of a graph $G$ is an \blue{edge $e$} such that + \[ + c(G - \blue{e}) > c(G). + \] + \end{definition} \end{frame} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \begin{frame}{} + \begin{theorem}[Characterization of Trees] + \begin{enumerate}[(1)] + \setcounter{enumi}{2} + \setlength{\itemsep}{6pt} + \item $T$ is connected, and has $n-1$ edges; + \item $T$ is connected, and each edge is a \cyan{bridge}. + \end{enumerate} + \end{theorem} + + \pause + \vspace{0.30cm} + \begin{center} + Consider any edge \blue{$e$} of $T$. + \end{center} + + \pause + \vspace{-0.20cm} + \[ + m(T - \blue{e}) = (n - 1) - 1 = n - 2. + \] + + \pause + \vspace{-0.20cm} + \begin{center} + $T - \blue{e}$ must be disconnected. + \end{center} \end{frame} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \begin{frame}{} + \begin{theorem}[Characterization of Trees] + \begin{enumerate}[(1)] + \setcounter{enumi}{3} + \setlength{\itemsep}{6pt} + \item $T$ is connected, and each edge is a \cyan{bridge}; + \item Any two vertices of $T$ are connected by exactly one path. + \end{enumerate} + \end{theorem} + + \pause + \vspace{0.30cm} + \begin{center} + Consider any two vertices $u$ and $v$. + + \pause + \vspace{0.30cm} + $T$ is connected $\implies$ $u$ and $v$ are connected by $\ge 1$ path. + + \pause + \vspace{0.60cm} + \red{If $u$ and $v$ are connected by two paths}, \\[3pt] + the edges on these two paths are not bridges. + \end{center} \end{frame} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% \begin{frame}{} + \begin{theorem}[Characterization of Trees] + \begin{enumerate}[(1)] + \setcounter{enumi}{4} + \setlength{\itemsep}{6pt} + \item Any two vertices of $T$ are connected by exactly one path; + \item $T$ is acyclic, but the addition of any edge creates exactly one cycle. + \end{enumerate} + \end{theorem} + + \pause + \vspace{0.30cm} + \begin{center} + \red{If $T$ has a cycle $C$}, \\[3pt] + any two vertices in $C$ is connected by $\ge 2$ paths. + + \pause + \vspace{0.60cm} + Consider the addition of edge $\set{u, v}$ to $T$. \\[3pt] + It creates a cycle, consisting of $\set{u, v}$ and the path from $u$ to $v$. + + \pause + \begin{lemma} + If two distinct cycles of a graph $G$ share \blue{a common edge $e$}, \\[3pt] + then $G$ has a cycle that does \purple{not} contain $e$. + \end{lemma} + \end{center} \end{frame} %%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%% +\begin{frame}{} + \begin{theorem}[Characterization of Trees] + \begin{enumerate}[(1)] + \setcounter{enumi}{5} + \setlength{\itemsep}{6pt} + \item $T$ is acyclic, but the addition of any edge creates exactly one cycle; + \setcounter{enumi}{0} + \item $T$ is a tree. + \end{enumerate} + \end{theorem} + + \pause + \vspace{0.30cm} + \begin{center} + \red{Suppose that $T$ is disconnected.} + + \pause + \vspace{0.60cm} + $T$ is a forest, consisting of $\ge 2$ trees $T_{1}, T_{2}, \dots$ + + \pause + \vspace{0.30cm} + Choose $u \in V(T_{1})$, $v \in V(T_{2})$. + + \vspace{0.30cm} + $T + \set{u, v}$ does \red{not} create cycles. + \end{center} +\end{frame} +%%%%%%%%%%%%%%% \ No newline at end of file