D1 - Matchings

Hello,
I'm afraid that I've suddenly become of those people who really used to annoy me when I tried to help them, because I had my first lesson on graphs and matching today in Further Maths, and I don't understand any of it. The entire concept escapes me, even though there doesn't seem to be anything complicated involved. My teacher has tried multiple times to help me, and I've looked up definitions for myself, but it still just feels like reading words and not understanding a thing. Could anyone please explain and define matching, maximal matching, maximum matching and perfect matching for me, in very simple language?

As an example, here is a definition that doesn't mean anything to me: "A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it is no longer a matching, that is, M is maximal if it is not a proper subset of any other matching in graph G. In other words, a matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M."

Sorry for the long post, but I'm trying to make the questions as clear as possible.

Thanks.

TL;DR Could anyone please explain and define matching, maximal matching, maximum matching and perfect matching for me, in very simple language?

I know what you mean!

In English:
You have two sets of vertices (dots) and a graph of edges (lines) joining them. A matching is some of those edges, but there can't be more than 1 edge going to any 1 vertex. Some vertices may not have any edges going to them.

The trivial matching is one where there are no edges.

A maximal/maximum matching is one with as many edges as possible. For example, if there are 3 vertices in each set, it will often be possible to find 3 edges in the matching, so that will be maximum. If one set has 3 vertices and the other has 4, then you can't have more than 3 edges in the matching. Depending on the original graph, it may not be possible to get that many edges in the matching.

A complete (or perfect?) matching is one where every vertex has an edge going from it. It will not be possible to find a complete matching if the two sets have different numbers of vertices.

Story time: We are trying to sort out who runs which stall at the village fete. If person A is the only one who is prepared to run the book stall and the only one who is prepared to run the tombola, one of those stalls can't happen so it will not be possible to get a complete matching, even if you have the same number of people as you do stalls.

Ohhhh.... thanks! This is about a million times clearer than anything else I've seen (and thanks for the example too, it was really helpful). I think I can even take on the homework now. In any case I'll print what you said and add it to my folder because it will be a whole lot more useful to me than today's class notes were.

Thanks a million for taking the time to put everything in idiot proof language, I owe you one.