# minimum number of edges that should be added to an undirected graph to make it a tree

Your first guess is correct. Sometimes there is more than one way to write the same solution.

Clearly, if there are $$k$$ connected components you’ll need exactly $$k-1$$ edges to connect them (without forming any cycle).

On the other hand, a tree with $$n$$ nodes must have exactly $$n-1$$ edges, so if the graph is acyclic and already has $$m$$ edges, then it is missing $$n-1-m$$ edges.

Regarding the example: the given graph has $$n=10$$ vertices, $$m=6$$ edges, and $$k=4$$ connected components (not 3), so the answer is $$3=k-1=n-m-1$$. The sets of vertices in each connected component are $${1,2,8}, {3}, {4, 6, 10}$$, and $${5, 7, 9}$$.