Two definitions of Safe Edge I ran into an interview two days ago and came across one strange definition of safe edge.
We are given an undirected weighted Graph $G = (V,E)$ with all distinct edge weights. Assume that the graph is connected.

Safe Edge Definition: if an edge $e \in E$ is not contained in any cycle, we called it a safe edge

Following is a theorem based on the above definition.

Theorem: All the safe edges must be in Minimum Spanning Tree (MST).

Please compare the definition of a &quot;safe edge&quot; on this interview with the classical definition of safe edge at all courses like here: MAIN SAFE EDGE THEOREM.
They are two different definitions.
The interview definition calls a safe edge one that is not part of any cycle and therefore cannot be removed from $G$ without disconnecting it, thus changing the resulting MST.
Notice that this definition depends solely on the chosen edge and on $G$. Notice also that it does not depend on the edge weights in any way.
The definition in the notes defines a safe edge $e$ as one that can safely be added to a subgraph $A$ of $G$ so that $A + e$ is a partial MST of $G$.
In this case $A$ is probably thought to already be a partial MST of $G$, since otherwise no edge would be safe, although this is not formally required.
Notice that this definition depends on the particular choice of $A$ and on the edge weights of $G$.
As an example, let $G$ be a cycle on $n$ nodes, where the $n$ edges have weights $1, \dots, n$. None of these edges is safe according to the interview definition.
If $A$ is the empty graph then all edges except for the one of weight $n$ are safe w.r.t. the definition in the notes.
If $A$ is the subgraph containing the edge of weight $n$ then no edge is safe w.r.t. the definition in the notes.