Here is an outline: Lemma:Consider a connected graph with its vertices are some vertices of the table and edges some edges of the grid.Let $a$ be the number of vertices and $d$ the number of edges.Then $d \leq 2(a-\sqrt{a})$. Proof:Suppose this vertices lye in $x$ rows and $y$ rows.Now,denote $xi$ the number of vertices in $i$-th row ,and $yi$ the number of vertices in $i$-th column.Now,it is obvios that if we have $m$ vertices in some row/column,we can have at most $m-1$ edges that belong to that row,so sum this over all rows and columns and we obtain that we can have maximum $2a-x-y$ and since $xy>=a$ it means that $x+y$ is minimum $2\sqrt{a}$(by $AM-GM$),so we proved our lemma. Now,suppose we have $c$ connected components and that component $i$ has $ci$ edges.Just use the inequality from the above lemma, sum over all and we proved that the minimum is at least $l/2$.Now,it remains to find an example which is easy from the above lemma( consider a grid $k*k$ or $k(k+1)$ and connect mostly all edges,I will write in detail if neccesary but it is easy).