Originally Posted by
Plato This is again one of those situations in which I don’t for sure what you have to use. In general a tree has one less edge than vertices. If a component (btw. saying connect is redundant components are connected) has a cycle then it has at least as many edges as vertices. That would contradict the given that |E|=|V|-p.
If every component is a tree then $\displaystyle \left| E \right| = \sum\limits_{k = 1}^p {\left| {E_k } \right|} = \sum\limits_{k = 1}^p {\left( {\left| {V_k } \right| - 1} \right)} = \left| V \right| - p$.
So a cycle any where changes that equation.