A graph $G$ is \textit{arbitrarily partitionable} (AP for short) if for every partition $(\tau_1, ..., \tau_p)$ of $|V(G)|$ there exists a partition $(V_1, ..., V_p)$ of $V(G)$ such that each $V_i$ induces a connected subgraph of $G$ with order $\tau_i$. If, additionally, $k$ of these subgraphs ($k\le p$) each contains an arbitrary vertex of $G$ prescribed beforehand, then $G$ is \textit{arbitrarily partitionable under $k$ prescriptions} (AP+$k$ for short). Every AP+$k$-graph on $n$ vertices must be $(k+1)$-connected, and have thus at least $\lceil \frac{n(k+1)}{2} \rceil$ edges. We show that there exist AP+$k$-graphs on $n$ vertices and $\lceil \frac{n(k+1)}{2} \rceil$ edges for every $k \geq 1$ and $n \geq k$.