Let $\Sigma$ be an oriented, compact, connected 2-manifold with boundary. Assume that its boundary is equipped with a disjoint union decomposition into two non-empty parts:
$$\partial\Sigma=\partial_{in}\Sigma\cup\partial_{in}\Sigma$$
(both $\partial_{in}\Sigma$ and $\partial_{in}\Sigma$ are disjoint unions of circles).<br>

> The following should be well known, and I'm looking for a reference:

> $\bullet\quad$ $\Sigma$ admits a relative handle decomposition that only uses $1$-handles (no need to use $0$-handles or $2$-handles).

> $\bullet\quad$ One can go from any such handle decompositions to any other one by (1) isotopies (2) handle slides (no need to ever introduce $0$-handles or $2$-handles).

If someone has a short and elegant proof of the above result, that would also make me happy.