# Where can I find out more about the nature of holes in plane regions?

Over the last few months I have been studying tilings of regions by polyominoes (mostly dominoes). I have been putting my findings together, mostly in the form of proving various things about polyomino regions and their tilings.

But a few times now I had to gloss over some details, all which seem to me related. These seem obvious, but I would like to have a list of properties / facts that I can rely on to reason more confidently. Here are some examples of the types of things I am interested in:

• Distinguishing the outside border from the borders of holes.
• If we have polyomino region that can be extended arbitrarily at specified edges on the outside border, without any new cells neighboring cells of the original region except at the specified cells, then
• the extensions can connect with each other only on specific ways without overlap (for example, if we have four extendable edges 0, 1, 2, 3 in clockwise order, and the extensions at 0 and 2 connect somewhere outside the original region, then the extensions at 1, and 3 cannot also connect outside the figure)
• if extensions connect up in a certain way, it forces some holes to form.
• Holes are simply-connected.
• One hole cannot surround another in a connected region (or overlap).

Part of it has to do with choosing a suitable definition of polyomino, and then a suitable definition of hole. (In many papers that deal with polyominoes with holes, a more complicated definition is used for polyomino than is necessary for most other purposes.)

Some of the types of things where these issues arise can be seen from some of my other questions here, for example: Is every “even” polyomino with one hole tileable by dominoes? , Elementary proof of transformations of domino tilings.. Another example is the following formula that applies to polyomino regions:

$$h = \frac{v - p}{4} +1,$$

where $h$ is the number of holes, $v$ is the number of valleys (sides of the polyomino between concave corners), $p$ is the number of peaks (sides of the polyomino between convex corners).

So my question is, where can I find out more about how holes in plane regions (maybe specifically polyomino regions) really work?

Ideally, I would not like to delve to deep into too abstract topics, so the more elementary the better.

(I do not really know which are suitable tags to apply. The ones I added seem relevant, but please suggest alternatives if they apply.)