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Topic: An Argument for the Subroot Node
Replies: 28   Last Post: Mar 14, 2008 8:32 PM

 Messages: [ Previous | Next ]
 Tony Orlow Posts: 3,142 Registered: 8/23/06
Re: An Argument for the Subroot Node
Posted: Mar 11, 2008 1:23 PM

G. Frege wrote:
> On Tue, 11 Mar 2008 08:40:29 -0700 (PDT), Tonico <Tonicopm@yahoo.com>
> wrote:
>

>>> "In graph theory, a tree is a graph in which any two vertices are
>>> connected by /exactly one/ path."
>>>

>> Where did you get that definition from?
>>

> Wikipedia. Entry "Tree (graph theory)".
>

>> According to it, the perimeter of a triangle is a tree...
>> which, of course, it is not.
>>

> Wait a second. If the nodes of your triangle were, say a, b, c then
> (say) a, b would be connected by more than one path:
>
> p_1 = ((a,b), (b,c), (c,a))
>
> p_2 = ((a,b), (b,c), (c,a), (a,b), (b,c), (c,a))
> [ or
> p_3 = ((a,c), (c,b), (b,a)) ]
>
> See?
>

Yes, you're right about that. There are two simple paths between a and
b. It's a cycle, for that reason.

Even in a tree, there are multiple paths between any two nodes, if one
is allowed to repeat nodes. Clearly that's not allowed, so a tree is
best defined as having exactly one *simple* path, without any nodes
repeated, between any pair of nodes. Given this restriction, little
self-directed loops don't affect that definition, since they are not
allowed in any simple path due to the fact that they repeat a node. They
are not true cycles, since they do not introduce new simple paths within
the tree. They can be appended without damaging the integrity of the
tree. They are not generally useful when scattered about, but the
subroot node is justified, and not proscribed by the above definition.

See?

>> A tree is a connected graph without any cycles. Period. This is not an
>> "alternative" definition to the nonsense written above.
>>

> Well... sure? See my (part of an) "argument" from above.
>
>
> F.
>

T.

Date Subject Author
3/11/08 Tony Orlow
3/11/08 Virgil
3/11/08 G. Frege
3/11/08 Tony Orlow
3/11/08 Virgil
3/11/08 Tony Orlow
3/11/08 Virgil
3/11/08 G. Frege
3/11/08 J. Antonio Perez M.
3/11/08 G. Frege
3/11/08 G. Frege
3/11/08 Tony Orlow
3/11/08 briggs@encompasserve.org
3/14/08 Tony Orlow
3/14/08 David R Tribble
3/14/08 LudovicoVan
3/11/08 Tony Orlow
3/11/08 J. Antonio Perez M.
3/11/08 Tony Orlow
3/11/08 Tony Orlow
3/11/08 Tony Orlow
3/11/08 LudovicoVan
3/11/08 LudovicoVan
3/14/08 Tony Orlow
3/14/08 LudovicoVan
3/14/08 LudovicoVan
3/11/08 David R Tribble
3/11/08 LudovicoVan
3/11/08 LudovicoVan