Suppose:
$x_1,x_2,x_3,x_4$ is a shortest path from $x_1$ to $x_4$
$x_2,x_5,x_6,x_7,x_9$ is a shortest path from $x_2$ to $x_9$
$x_{10},x_5,x_8,x_3,x_9$ is a shortest path from $x_{10}$ to $x_9$
Which of the following is/are true?
(a) $x_1,x_2,x_5,x_6,x_7,x_9$ is a shortest path from $x_1$ to $x_9$
(b) The length of $x_5,x_6,x_7,x_9$ must be the same as $x_5,x_8,x_3,x_9$
(c) At least one edge in the graph must have 0 length
I'm confused as to how to work out these conceptually. I thought (a) was true, but it turns out that it's false. I don't understand why (b) must be true, nor why (c) must be false. If anyone can outline a procedure that will help me with this problem, and just understanding the shortest path in general, I'd greatly appreciate it.
The solution says that, for (b), "the subpath of the shortest path is also a shorest path." Can someone explain what this means, and how I can recognize that this is actually true?
Thank you.