= Solution
For the example, take a once-punctured <torus> $P$, an essential nonseparating simple closed curve $c\subset P$, and
$$
M=P\times I,\qquad A=c\times I.
$$
The inclusion of the <annulus> $A$ induces the <injection> $\langle c\rangle\hookrightarrow\pi_1(P)$, so $A$ is an <incompressible surface>. Cutting gives $(P\text{ cut along }c)\times I$, a <pair of pants> times an interval. A <compact> <connected> <surface> with <boundary> times an interval is a <handlebody> of <genus> $1-\chi(P)$: cut the <surface> to a <disk> by proper arcs and reverse those cuts as one-handle attachments. Both the punctured <torus> and the pair of pants have <Euler characteristic> $-1$. Therefore
$$
\boxed{M\text{ and }M\text{ cut along }A\text{ are both genus-two handlebodies}.}
$$
This <self-reproducing incompressible annulus cut> also shows why arbitrary incompressible cutting choices need not terminate.
A <three-manifold hierarchy> is a finite sequence of cuts along properly embedded two-sided <incompressible surfaces>, with no spherical components, whose final manifold is a disjoint union of <three-balls>. <Disks> are permitted. In an <orientable> ambient manifold two-sided <surfaces> are <orientable>; for an ambient manifold which is not <orientable> we permit two-sided <surfaces> without assuming their <orientability> as well.
For $S^1\times F$, first cut along $\{p\}\times F$, obtaining $I\times F$. This <surface> is two-sided and its <fundamental group> injects as a direct factor. Next choose essential two-sided curves in $F$ until the cut <surface> has <boundary>, and cut along their products with $I$. For <orientable> $F$ one nonseparating curve suffices. For $F$ which is not <orientable> and has <genus> at least two, the <boundary> of an embedded <Möbius band> is a two-sided essential curve; neither side is a <disk>, so its <fundamental group> injects. Cutting this curve leaves pieces with <boundary>, including the <Möbius band>. For each resulting <compact> <surface> choose a finite system of proper arcs cutting it into <disks>. The corresponding vertical <disks> give the remaining <three-manifold hierarchy> cuts. Essential <annuli> have <injective> <fundamental group>, and <disks> have trivial <fundamental group>, so all cuts are incompressible. The terminal products are $D^2\times I$, hence <three-balls>. This construction covers the possibilities without <orientability> in the stated range as well as the <orientable> ones.
Here is a terminating construction for a <compact> <connected> <orientable> <irreducible three-manifold> $M$ with nonempty <boundary>. We use the permitted general results in the following precise forms: a boundary-incompressible non-ball component has a <connected> two-sided nonseparating <incompressible surface> with nonempty <boundary>; and a sufficiently large collection of disjoint closed <incompressible surfaces> has two parallel components with no intervening member. These are respectively the existence and finiteness results for <incompressible surfaces>.
First perform essential <boundary> <disk> compressions as long as possible. Their number is finite: each nonseparating compression decreases <boundary> <genus>, while a separating compression divides a <boundary> component into two of smaller <genus>. More explicitly, the sum of $\max(0,2g-1)$ over the closed boundary components decreases at every essential compression. Each cut preserves <three-manifold irreducibility>. To see this for any incompressible cutter, a <sphere> in the cut manifold bounds a <three-ball> in the uncut manifold. A closed nonspherical incompressible component of the cutter cannot lie in this <three-ball>, since its <fundamental group> could not inject into the trivial group. A component with <boundary> cannot lie in an interior <three-ball> either. Thus the <three-ball> misses the cutter and survives in the cut manifold. Fundamental-group <injectivity> of two-sided <incompressible surfaces> follows from the <Loop theorem>, applied to their two <boundary> copies after cutting.
Push the <boundary> of each non-ball residual component slightly into the original $M$ to make a closed frontier $F_1$, omitting spherical components. This frontier is incompressible on its residual side, since all <boundary> compressions there have been exhausted. It is also incompressible on the side already cut off. Indeed, a hypothetical compression <disk> on that side can be made transverse to the earlier cutting <surfaces>. An innermost intersection circle bounds a <disk> on its cutting <surface> by incompressibility; replacing the corresponding subdisk removes the circle, without introducing intersections with earlier cutters. Repeat in the order of the cuts. The <disk> would finally lie in a product collar of the frontier, where its <boundary> is fundamental-group-injective, a contradiction. Thus $F_1$ is incompressible in $M$.
If a residual component is not a <three-ball>, choose the permitted nonseparating <incompressible surface> with <boundary>, cut it, and compress the new <boundary> fully again. Push that <boundary> in to obtain a new closed frontier $F_2$ disjoint from $F_1$. Continue. <Three-manifold irreducibility> persists, and the same innermost-circle proof makes every nonspherical frontier incompressible in the original $M$.
We explain why the parallel frontiers forced by the finiteness theorem cannot occur in this construction. The required product fact is: \b[a <connected> nonseparating <incompressible surface> in $G\times I$ cannot be proper and disjoint from $G\times\{1\}$]. To prove it, intersect the <surface> with vertical <annuli> $c\times I$ for essential curves $c$ in a closed $G$. Inessential intersection circles can be removed using incompressibility and the <three-balls> supplied by <three-manifold irreducibility>. An outermost intersection arc has both endpoints on the lower <boundary>; <boundary> compression along its annular bigon reduces the intersections. At least one resulting component stays nonseparating, since the mod-two relative class of the original <surface> is the sum of the resulting classes. Essential intersection circles can likewise be removed by surgery along the <annulus> running to its lower <boundary>. Retain a nonseparating incompressible component after each surgery. These operations reduce the intersection number and cut the base <surface> to one with <boundary>. Now intersect with the vertical <disks> of a proper arc system cutting that base to a <disk>. The same innermost-circle and outermost-arc operations make the <surface> disjoint from these <disks>. It then lies in a <three-ball> $D^2\times I$, where any proper two-sided <surface> separates. This is the contradiction proving the product fact.
If two frontiers in the construction were parallel with no other frontier between, their product region would contain one of the intervening nonseparating cutters, disjoint from the later frontier. The product fact forbids this. Consequently the closed-frontier construction has only finitely many stages. Once a residual irreducible component has only spherical <boundary>, push one <boundary> <sphere> inward. The <three-ball> it bounds cannot be on the collar side, which contains a <boundary> component of the manifold. The <three-ball> and the collar therefore exhaust that component, making it a <three-ball>. We have constructed \b[a finite <three-manifold hierarchy>, and can choose all its cutting <surfaces> to have nonempty <boundary>].
We will use one further consequence in the map argument: these manifolds are <aspherical>. Work backwards through the <three-manifold hierarchy>. A <three-ball> is <contractible>. At one gluing stage, the <van Kampen theorem> normal form identifies the lifts of the cut pieces and cutting <surface> in the <universal cover> with a tree of spaces. The lifted pieces are <contractible> by induction, and the lifted cutting <surfaces> are <contractible> because the <surfaces> have <boundary> and are fundamental-group-injective. A finite subtree is <contractible> by removing terminal vertices; every <sphere> map meets only finitely many pieces. All higher <homotopy groups> therefore vanish at this stage, and induction proves <asphericity from a three-manifold hierarchy>. Apply this tree argument separately at each gluing stage.
The following three entries complete a single proof of the relative-boundary <homeomorphism> assertion.
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