I've been spending a lot of time reading online presentations and textbooks about the cut property of a minimum spanning tree. I don't really get what it's suppose to illustrate or even why it's practical. Supposedly it helps determine what edges to add to a MST, but I fail to see how it accomplishes that. My understanding of the cut property so far is that you split a MST into two arbitrary subsets. Any help here? Thanks!
A cut of a connected graph is a minimal set of edges whose removal separate the graph into two components (pieces). The minimal cut property says that if one of the edges of the cut has weight smaller than any other edge in the cut then it is in the MST. To see this, assume that there is an MST not containing the edge. If we add the edge to the MST we get a cycle that crosses the cut at least twice, so we can break the cycle by removing the other edge from the MST, thereby making a new tree with smaller weight, thereby contradicting the minimality of the MST.