\magnification 2100 \parindent 0pt \parskip 10pt \pageno=1 \hsize 7.2truein \hoffset -0.2truein \vsize 9.5truein \def\u{\vskip -10pt} \def\v{\vskip 2cm} If $G = (V, E)$ connected, $W \subset V$, $G - W$ disconnected, then $W$ {\it separates} $G$. If $s, t \in V - W$ and $s$ and $t$ in different components of $G - W$, then {\it $W$ separates $s$ from $t$}. \v If $k \geq 2$: $G = (V, E)$ is {\it $k$-connected} if removing any set of $k-1$ vertices in $G$ does not disconnect it and $|V| \geq k+1$ (or equivalently, $|V| \geq k+2$ or $G = K_{k+1}$). $G = (V, E)$ is {\it $k$-edge-connected} if removing any set of $k-1$ edges in $G$ does not disconnect it and $|V| \geq 2$. 1-connected = 1-edge connected = connected. $\kappa(G) = $ connectivity of $G$ = $max\{k ~|~ G ~k-connected \}$ $\kappa(G) = 0$ if $G$ disconnected. $\lambda(G) = $ edge connectivity of $G$ = $max\{k ~|~ G ~k-edge-connected \}$ $\lambda(G) - 1 \leq \lambda(G-xy) \leq \lambda(G)$ $\kappa(G) - 1 \leq \kappa(G - x)$ $ \kappa(G) \leq \lambda(G) \leq \delta(G)$ \v \vfill Recall, two $s-t$ paths are independent if they only have the vertices $s$ and $t$ in common. Thm 5 (Menger 1927) \hfil \break (i) Let $s, t$ distinct nonadjacent vertices in $G$. \hfil \break $min\{|W| ~|~ W \subset V(G), ~W $ separates $s$ from $t \}$ = maximum number of independent $s-t$ paths. (ii)Let $s, t$ distinct vertices in $G$. \hfil \break $min\{|E'| ~|~ E' \subset E(G), E' $ separates $s$ from $t \}$ = maximum number of edge-disjoint $s-t$ paths. Cor 6: For $k \geq 2$, $G$ is $k$-connected iff $V(G) \geq 2$ and any two vertices can be joined by $k$ independent paths. $G$ is $k$-edge-connected iff $V(G) \geq 2$ and any two vertices can be joined by $k$ edge disjoint paths. \eject If $G_1$, $G_2$ $k$-connected, $|V(G_1) \cap V(G_2)| \geq k$, then $G_1 \cup G_2$ is $k$-connected. Pf: If $|W| \leq k-1$, then $G_1 \cup G_2 - W = (G_1 - W) \cup (G_2 - W)$ \vskip 20pt A subgraph $B$ of $G$ is a {\it block of $G$} if $B$ is a bridge or if $B$ is a maximal 2-connected subgraph of $G$. If $B_1$ and $B_2$ are blocks, then $|V(B_1) \cap V(B_2)| \leq 1$. If $x,y \in V(B)$, a block, and $x \not= y$ then $G - E(B)$ does not contain an $x-y$ path. \v A vertex $v$ belongs to at least two blocks iff $v$ is a cutvertex. $E(G) = \cup_{i=1}^p E(B_i)$ where $E(B_i) \cap E(B_j) = \emptyset$ if $i \not= j$ and $B_i$'s are blocks. \eject Suppose $G$ nontrivial connected graph where $\{v_1, ..., v_n\}$ is the set of cutvertices of $G$.\hfil \break $bc(G) =$ block-cutvertex graph of $G$ = $(V', E')$ where $V' = \{v_1, ..., v_n, B_1, ..., B_p \}$ and $E' = \{(v_i, B_j) ~|~ v_i \in B_j \}$. $bc(G)$ is a tree. An endvertex of $bc(G)$ is a block of $G$ = {\it endblock} of $G$. \end \eject Slope of secant line between $(x_1, f(x_1))$ and $(x_2, f(x_2))$ \u \hskip 1in = average rate of change \u \hskip 1in = ${f(x_2) - f(x_1) \over x_2 - x_1}$ \u \hskip 1in = = ${\Delta f(x) \over \Delta x}$ \u where $\Delta x =$ change in $x$ = $x_2 - x_1$ \u and $\Delta f(x) = $ change in $f(x)$ = $f(x_2) - f(x_1)$ Slope of tangent line to $f$ at $x_1$ = instantaneous rate of change $$= lim_{x_2 \rightarrow x_1} {f(x_2) - f(x_1) \over x_2 - x_1}$$ $$= lim_{x_1 + h \rightarrow x_1} {f(x_1 + h) - f(x_1) \over x_1 + h - x_1}$$ $$ = lim_{h \rightarrow 0} {f(x_1 + h) - f(x_1) \over h}$$ Definition $f'(a) = $ slope of tangent line to $f$ at $a$ = $lim_{h \rightarrow 0} {f(a + h) - f(a) \over h}$. If $f(x) = 2x - 4$, then $f'(8) = $ If $g(x) = 3$, then $g'(1) = $ If $h(x) = |x|$, then $h'(5) = $ \hskip .8in and $h'(-5) = $ Definition: Given $f$, then define the function $f'$ (the derivative of $f$) as follows: \vskip -5pt $f'(x) = $ slope of tangent line to $f$ at $x$ = $lim_{h \rightarrow 0} {f(x + h) - f(x) \over h}$. $x$ is in the domain of $f'$ if $x$ is in the domain of $f$ and the above limit exists. If $f(x) = 2x - 4$, then $f'(x) = $ If $g(x) = 3$, then $g'(x) = $ If $h(x) = |x|$, then $h'(x) = $ Suppose $f(x) = -2x + 12$ represents the distance traveled from home in miles after $x$ hours. Find the average velocity between $x = 1$ and $x = 3$. What are the units? \v Find the instantaneous velocity at $x = 1$: What are the units? \v Suppose $f(x) = -2x + 12$ represents the cost of stock after $x$ days from purchase. Find the average change in the cost of the stock between $x = 1$ and $x = 3$. What are the units? \v Find the instantaneous change in the cost of the stock at $x = 1$: What are the units? \v Suppose $f(x) = 8$ represents the distance traveled from home in miles after $x$ hours. Find the average velocity between $x = 1$ and $x = 3$. What are the units? \v Find the instantaneous velocity at $x = 1$: What are the units? \v Suppose $f(x) = 8$ represents the cost of stock after $x$ days from purchase. Find the average change in the cost of the stock between $x = 1$ and $x = 3$. What are the units? \v Find the instantaneous change in the cost of the stock at $x = 1$: What are the units? \v Suppose $f(x) = {x + 3 \over 4x - 5}$ represents the distance traveled from home in miles after $x$ hours. Find the average velocity between $x = 1$ and $x = 3$. What are the units? \v Find the instantaneous velocity at $x = 1$: What are the units? \v Suppose $f(x) = {x + 3 \over 4x - 5}$ represents the cost of stock after $x$ days from purchase. Find the average change in the cost of the stock between $x = 1$ and $x = 3$. What are the units? \v Find the instantaneous change in the cost of the stock at $x = 1$: What are the units? \v \vfill HW \hfil \break 2.6) 1, 2, 3, 7, 9, 17, 21, 23, 29, 31, 33, 49a\hfil \break 2.7) 1, 2, 3, 9, 13, 15, 17, 27\hfil \break 2.8) 3, 4, 5, 7, 15, 25, 29, 33, 35, 36\hfil \break 2.9) 4, 7, 9, 21, 27, 29, 37, 38\hfil \break and \end no set of $k-1$ vertices in $V$ separates $G$.