1
$\begingroup$

It is a strange question on a book.

Give an example of a tree $T$ that does not satisfy the following property: If $v$ and $w$ are vertices in $T$, there is a unique path from $v$ to $w$.

I think it is rather strange because a tree $T$ is defined to be a simple graph and if $v$ and $w$ are vertices in $T$,there is a unique simple path from $v$ to $w$. Maybe the answer is the difference between "path" and "simple path"? Thanks for your help.

ps: It is the 39th exercise in 7.1 Exercises of the book Discrete Mathematics (Fifth Edition) written by Richard Johnsonbaugh.

  • 2
    Which book? $ $2012-04-07
  • 0
    either their trees can have several connected components (what we might consider a "forest") or, more likely (as you say), the paths are not simple2012-04-07
  • 0
    @Didier It is the 39th exercise in 7.1 Exercises of the book Discrete Mathematics (Fifth Edition) written by Richard Johnsonbaugh.Maybe it is a publishing mistake?2012-04-07
  • 0
    The author asks a tree to be connected hence the answer to your question is that a path can be not simple. For example the path 0-1-2-1-2-3 from 0 to 3 on the tree Z.2012-04-07
  • 1
    If the path is allowed to be non-simple, then *no* tree has the property that there is a unique path joining each pair of vertices...2012-04-07
  • 0
    @Didier Anyway,this question is rather strange!If a path can be not simple,then any tree having more than one vertex is the answer,just as Mariano says.2012-04-07
  • 0
    @MarianoSuárez-Alvarez: There's always the empty tree.2012-04-07
  • 0
    Sensible people will say that the empty tree is not connected, though! :D2012-04-07
  • 0
    @Mariano: The empty tree IS connected since one cannot find a partition of its vertex set into two nonempty subsets such that blablabla.2012-04-07
  • 0
    I know, I know... But many very sensible sources will exclude it from the definition, like one decides that the zero modules is not simple/indecomposable or that the empty space is not connected. This was amply discussed in MO a while ago.2012-04-07
  • 0
    Are $v$ and $w$ given to be _distinct_ vertices?2012-04-13
  • 0
    @DougChatham I don't know.It doesn't specify this.I think this exercise is a bit of ambiguous.2012-04-13

0 Answers 0