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| ``connected component''
by marijke on 2005-04-05 13:52:03 |
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| In my correction (about not every subset that's connected being a connected *component*) i suggested added something along the lines of
> Let $X\sim Y$ denote there exists a path joining vertices X and Y;
> it is not hard to see $\sim$ is an equivalence relation. Each of
> its equivalence classes, together with the edges connected to the
> vertices in that class, is known as a (\emph{connected})
> \emph{component} of the graph.
>
> A \emph{connected graph} is one that consists of a single connected
> component.
I'm glad you didn't take this suggestion up (yet) because my use of the $\sim$ symbol is unfortunate. That notation is already in use for "are linked directly by an edge", not a transitive concept. For the (transitive) relation on vertices of being connected by a path, any other ad hoc symbol would do in the course of the argument, such as $\approx$.
--regards, marijke
http://web.mat.bham.ac.uk/marijke/ |
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