116 lines
2.2 KiB
TeX
116 lines
2.2 KiB
TeX
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\documentclass[12pt]{article}
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\usepackage{ntnu}
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\usepackage{ntnu-math}
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\author{Øystein Tveit}
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\title{MA0301 Exercise 10}
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\usepackage{amsthm}
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\usepackage{mathabx}
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\usetikzlibrary{arrows.meta}
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\begin{document}
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\ntnuTitle{}
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\break{}
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Because there are no exercise where there are multiple edges between two vertices, I will use strings of vertex names to represent a walk.
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\begin{excs}
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\exc{}
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\begin{subexcs}
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\subexc{}
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\[ bcbcd \]
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\subexc{}
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\[ bacbed \]
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\subexc{}
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\[ bcd \]
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\subexc{}
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\[ bcb \]
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\subexc{}
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\[ befgedcb \]
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\subexc{}
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\[ bacb \]
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\end{subexcs}
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\exc{}
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\begin{figure}[H]
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\center
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\scalebox{2}{
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\input{diagrams/ex2.tex}
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}
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\end{figure}
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\exc{}
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By using trial and error, starting with the nodes that had a higher degree, I managed to bring it down to three nodes.
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\includeDiagram[scale=2, width=12cm]{diagrams/ex3.tex}
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The red vertices represent the guards
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\exc{}
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\begin{subexcs}
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\subexc{}
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The graphs are not isomorphic because the shortest cycle between the vertices with a degree of 3 has a different length.
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\subexc{}
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The graphs are isomorphic
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\end{subexcs}
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\exc{}
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\begin{subexcs}
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\subexc{}
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\includeDiagram[scale=2, width=12cm]{diagrams/ex5_a.tex}
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\[ adhijkgcbgjfbefiedba \]
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\subexc{}
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Because $deg(e) = deg(f) = 3$ is now odd, they have to be the starting vertex and ending vertex.
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\includeDiagram[scale=2, width=12cm]{diagrams/ex5_b.tex}
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\[ dabdhijkgcbgjfbefie \]
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\end{subexcs}
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\exc{}
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\begin{subexcs}
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\subexc{}
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$G_1$ is not an induced subgraph if it's missing an edge $e_1$ between $v_1, v_2 \in G_1$ where $e_1 \in G$
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\subexc{}
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\includeDiagram[scale=0.8, width=6cm, pdf=true]{diagrams/ex6_b.pdf}
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$G_1$ contains the vertices $c$ and $d$ while it is missing the edge $cd$ even though $cd$ was present in $G$. Therefore, it is not an induced subgraph
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\end{subexcs}
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\exc{}
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\begin{align*}
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\sum_{deg(v) \in V} = 2 |E| \\
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3|V| \leq 2 |E| \\
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|V| \leq \frac{2 |E|}{3} \\
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|V| \leq \frac{2 \cdot 17}{3} \\
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|V| \leq \frac{34}{3} \\
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|V| \leq 11.33 \\
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\end{align*}
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The max amount of vertices in $G$ has to be $11$
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\end{excs}
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\end{document}
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