Path decomposition

Path decomposition is a way of breaking up a really long path into smaller pieces. Imagine you have a really big puzzle, and you want to make it easier to put together. You could break it up into smaller parts and work on each part one by one, right? Path decomposition is kind of like that.

So let's say you have a path that looks like this: 1 - 2 - 3 - 4 - 5 - 6 - 7. That's a lot of numbers! It might be hard to work with that whole path at once. But if we break it up into smaller pieces, it's easier to handle.

One way to do that is to divide it into pairs of adjacent vertices. So we could break it up like this: 1 - 2, 2 - 3, 3 - 4, 4 - 5, 5 - 6, 6 - 7. Each of those little pairs is called a "segment".

Now we can work on each segment one by one. Maybe we want to color each segment a different color, or maybe we want to find the shortest path from one end of the path to the other. It's much easier to do that with these smaller, more manageable segments than with the whole long path at once.

That's the basic idea of path decomposition! It's just a way of breaking up a long path into smaller pieces so we can work with it more easily.