I'm studying for a graph theory exam. And while looking at some exams of previous years which were handed out by the professor, I found the following question.
As I understand it, a maximal matching is a matching with the highest possible weight, but this graph doesn't display weights.
The next part of the question, not shown in the image, asks for a maximum matching, given the initial matching from (a), which I know how to construct using augmenting paths.
The important word here is maximal (that is the reason that the 'al' is underlined). You do not want a matching of maximum size. This could not have size 3 because for example the edges ag, bh, ci, and df are independent.
The idea in this exercise is, that you want to find 3 edges that are independent (and thus form a matching) but at the same time all other edges need to be incident with at least one of those 3.
In the picture the vertices e and j are cut off (I suppose), if they are adjacent, the 3 edges bg, di, and ej seem to be a maximal matching:
They clearly form an independent set and each of a, c, f, and h has neighbours only in {b, d, e, g, i, j}.