Relation Between Bridges And Spanning Trees

= Bridges and spanning trees = Let $G$ be a simple graph and $T$ a spanning tree for $G$.

Theorem
An edge $e$ is a bridge in $G$ $\iff$ $e$ belongs to every spanning tree $T$

Necessary Condition
If $e$ is a bridge then it belongs, by definition, to every path between a pair of vertices $u$ and $v$. Therefore $e$ has to be in $T$ in order to be a spanning tree of $G$, otherwise $u$ and $v$ would not be connected in $T$.

Sufficient Condition
= Bridges and minimum spanning trees = Let $G$ be a simple weighted graph and $T$ a minimum spanning tree for $G$. Lets assume every edge has a unique weight associated to it.

Theorem (minimum weighted bridge)
An edge $e$ is a bridge of minimum weight in $G$ $\iff$ $e$ belongs to every minimum spanning tree $T$

Necessary Condition
Assume the contrary, $e$ is a bridge of minimum weight but it doesnt belong to some minimum spanning tree $Q$. Adding $e$ to $Q$ creates an unique cycle $C$ in $Q$ but then there exists $v$ $\in$ $C$ such that $w(Q)$ < $w(Q+e-v)$ contadicting its minimality.

Theorem (maximum weighted bridge)
An edge $e$ belongs to every minimum spanning tree $T$ for $G$ and has maximum weight in $G$ $\implies$ $e$ is a bridge in $G$