Is the path between 2 vertices of a Minimum weight tree of a graph the shortest path between those 2 vertices?

Question

Suppose we have an undirected, connected graph, $G_1$

If you have a minimum weight spanning tree $G_2$ for graph $G_1$. Is it possible to find two vertices in $G_1$ which is has a shortest path that is not present in $G_2$?

Intuitively it seems that this is not possible otherwise that shortest path would be part of $G_2$ but I would like to know if I am wrong

Answer 1

Yes. Consider the two vertices which have maximum distance in the spanning tree. It is not hard to see that the two vertices can be connected with an edge with weight low enough to be less that the weight of the path, but high enough to be excluded in the formation of tree.

Consider the graph as below, where W(m, n) is the weight of the edge connecting edges m and n:

W(1,2)=1, W(1,3)=2, W(3,4)=3, W(1,4)=4.

The minimal spanning tree consists of W(1,2),W(1,3),W(3,4), while path between vertices 1 and 4 has weight 4 in the graph but weight 5 in the spanning tree. Hope this helps.

Answer 2

HINT:

Think of a triangle $ABC$ with $AB=3$, $BC=4$, $AC=5$. The minimum spanning tree contains the edges $AB$, $BC$ and $AB+ BC > AC$.