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undirected_graph [2014/02/09 20:51] nikolaj |
undirected_graph [2014/06/18 15:40] nikolaj |
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| ===== Undirected graph ===== | ===== Undirected graph ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $V,E$ ... set | | + | | @#55CCEE: context | @#55CCEE: $V,E$ ... set | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $ \langle V,\langle E,\psi\rangle\rangle \in \mathrm{it}(E,V) $ | | |
| - | | @#FFBB00: $ \langle V,\langle E,\psi\rangle\rangle \in \mathrm{it}(E,V) $ | | + | | @#55EE55: postulate | @#55EE55: $ \psi $ ... function | |
| - | + | | @#55EE55: postulate | @#55EE55: $ \mathrm{dom}(\psi)=E $ | | |
| - | | @#55EE55: $ \psi $ ... function | | + | | @#55EE55: postulate | @#55EE55: $ \forall (e\in E).\ \exists (u,v\in V).\ \psi(e) = \{v,u\} $ | |
| - | | @#55EE55: $ \mathrm{dom}(\psi)=E $ | | + | |
| - | + | ||
| - | | @#55EE55: $ \forall (e\in E).\ \exists (u,v\in V).\ \psi(e) = \{v,u\} $ | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| In the above definition, the set $E=\{a,b,\dots\}$ in $\langle E,\psi\rangle$ is any set whos elements then each label an edge, e.g. $\psi(a)=\{v,w\}$. | In the above definition, the set $E=\{a,b,\dots\}$ in $\langle E,\psi\rangle$ is any set whos elements then each label an edge, e.g. $\psi(a)=\{v,w\}$. | ||
| - | Instead, one can also define a graph using a [[multiset]] $\langle E_\mathrm{ends},m\rangle$ where $E_\mathrm{ends}=\{\{v,w\},\{u,w\},\dots\}$ is itself a set of endpoints and $m:E_\mathrm{ends}\to\mathbb N$ counts the number of instances such a pair is part of the graph. The definitions are of course practically equivalent, but the definition with $\psi$ de-emphasises the focus on "$v$ and $w$ from $V$ are things which are connected" in favor of "$a$ is something from $E$ which connects the things $v$ and $w$ from $V$". | + | Instead, one can also define a graph using a [[multiset]] $\langle E_\mathrm{ends},m\rangle$ where $E_\mathrm{ends}=\{\{v,w\},\{u,w\},\dots\}$ is itself a set of endpoints and $m:E_\mathrm{ends}\to\mathbb N$ counts the number of instances such a pair is part of the graph. The definitions are of course practically equivalent, the definition above with $\psi$ de-emphasises the focus on "$v$ and $w$ from $V$ are things which are connected" in favor of "$a$ is something from $E$ which connects the things $v$ and $w$ from $V$". |
| ==== Parents ==== | ==== Parents ==== | ||
| === Subset of === | === Subset of === | ||
| [[Graph]] | [[Graph]] | ||
| - | === Requirements === | + | === Context === |
| [[Function]] | [[Function]] | ||
