Question 104
Consider an $n \times n$ array of points. For a subset $A$ of the edges of this array (i.e. the lattice edges), we denote by $V(A)$ the set of vertices of $A$ and by $J(A)$ the set of connected components of $A$. For each natural number $l$, prove that
$$
\frac{l}{2} \le \min_{|A| \ge l} (|V(A)| - |J(A)|) \le \frac{l}{2} + \sqrt{\frac{l}{2}} + 1.
$$
(For a set $X$, by $|X|$ we mean the number of elements of $X$).
$$
\frac{l}{2} \le \min_{|A| \ge l} (|V(A)| - |J(A)|) \le \frac{l}{2} + \sqrt{\frac{l}{2}} + 1.
$$
(For a set $X$, by $|X|$ we mean the number of elements of $X$).