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Asked: 2026-06-03T05:53:00+00:00

Here is an exercise: Either prove the following or give a counterexample: (a) Is

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Here is an exercise:

Either prove the following or give a counterexample:

(a) Is the path between a pair of vertices in a minimum spanning tree
of an undirected graph necessarily the shortest (minimum weight) path?

(b) Suppose that the minimum spanning tree of the graph is unique. Is
the path between a pair of vertices in a minimum spanning tree of an
undirected graph necessarily the shortest (minimum weight) path?

My Answer is

(a)

No, for example, for graph 0, 1, 2, 0-1 is 4, 1-2 is 2, 2-0 is 5, then 0-2’s true shortest path is 5, but the mst is 0-1-2, in mst, 0-2 is 6

(b)

My problem comes into this (b).

I don’t understand how whether the MST is unique can affect the shortest path.

First, my understanding is that when the weights of edges are not distinct, multiple MST may exist at the same time, right?

Second, even if MST is unique, the answer of (a) above still applies for (b), right?

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1 Answer

  1. Editorial Team

    Regarding (a), I agree.

    Regarding (b), for some graphs, there may be more minimal spanning trees with the same weight. Consider a circle C3 with vertices a,b,c; weights a->b=1, b->c=2, a->c=2. This graph has two MSTs, {a-b-c} and {c-a-b}.

    Nevertheless, your counterexample still holds, because the MST is unique there.

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