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70 lines
2.3 KiB
TeX
70 lines
2.3 KiB
TeX
\subsection{Zusammenhang von Graphen: Was ist das?}
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\begin{frame}{Zusammenhang von Graphen}{Connectivity}
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\begin{block}{Streng zusammenhängender Graph}
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Ein streng zusammenhängender Graph ist ein gerichteter Graph,
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in dem jeder Knoten von jedem erreichbar ist.
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\end{block}
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\begin{figure}
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\begin{tikzpicture}[->,scale=1.8, auto,swap]
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% Draw a 7,11 network
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% First we draw the vertices
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\foreach \pos/\name in {{(0,0)/a}, {(0,2)/b}, {(1,2)/c},
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{(1,0)/d}, {(2,1)/e}, {(3,1)/f},
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{(3,2)/g}, {(2,0)/h}}
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\node[vertex] (\name) at \pos {$\name$};
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% Connect vertices with edges and draw weights
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\foreach \source/ \dest /\pos in {a/b/,b/c/,c/d/,d/a/,
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c/e/bend left, d/e/,e/c/, f/g/,
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g/f/bend left, d/h/}
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\path (\source) edge [\pos] node {} (\dest);
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\end{tikzpicture}
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\end{figure}
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\end{frame}
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\begin{frame}{Zusammenhang von Graphen}{Connectivity}
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\begin{block}{Zusammenhangskomponente}
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Eine Zusammenhangskomponente ist ein maximaler Subgraph S
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eines gerichteten Graphen, wobei S streng zusammenhängend ist.
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% Muss dieser Subgraph maximal sein?
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\end{block}
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\begin{figure}
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\begin{tikzpicture}[->,scale=1.8, auto,swap]
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% Draw a 7,11 network
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% First we draw the vertices
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\foreach \pos/\name in {{(0,0)/a}, {(0,2)/b}, {(1,2)/c},
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{(1,0)/d}, {(2,1)/e}, {(3,1)/f},
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{(3,2)/g}, {(2,0)/h}}
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\node[vertex] (\name) at \pos {$\name$};
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% Connect vertices with edges
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\foreach \source/ \dest /\pos in {a/b/,b/c/,c/d/,d/a/,
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c/e/bend left, d/e/,e/c/, f/g/,
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g/f/bend left, d/h/}
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\path (\source) edge [\pos] node {} (\dest);
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% colorize the vertices
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\foreach \vertex in {a,b,c,d,e}
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\path node[selected vertex] at (\vertex) {$\vertex$};
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\foreach \vertex in {f,g}
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\path node[blue vertex] at (\vertex) {$\vertex$};
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\foreach \vertex in {h}
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\path node[yellow vertex] at (\vertex) {$\vertex$};
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\end{tikzpicture}
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\end{figure}
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\end{frame}
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\begin{frame}{Elementare Eigenschaften}
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\begin{block}{}
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Die Knotenmengen verschiedener SCCs sind disjunkt.
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\end{block}
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\begin{block}{}
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SCCs bilden Zyklen.
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\end{block}
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\begin{block}{}
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Die Vereinigung aller Knoten aller SCCs ergibt alle Knoten
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des ursprünglichen Graphen.
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\end{block}
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\end{frame}
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