Edge is Minimum Weight Bridge iff in All Minimum Spanning Trees

Theorem

Let $G$ be an undirected network.

Let every edge of $G$ have a unique weight.

Let $e$ be an edge of $G$.

Then $e$ is a bridge of minimum weight in $G$ if and only if $e$ belongs to every minimum spanning tree of $G$.

Proof

Necessary Condition

Aiming for a contradiction, suppose $e$ is a bridge of minimum weight that does not belong to some minimum spanning tree $Q$.

Let $e$ be added to $Q$ to make $Q'$.

Then $e$ forms part of a unique cycle $C$ in $Q$.

Thus there exists an edge $f \in C$ such that $\map w Q < \map w {Q + e - f}$.

This contradicts the minimality of $Q$.

$\Box$

This article contains statements that are justified by handwavery. In particular: The above argument (cleaned up considerably from the original, which was appallingly inadequate) needs to be made rigorous. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding precise reasons why such statements hold. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Handwaving}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sufficient Condition

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.