adjacency matching in an undirected graph

Question

I am having trouble understanding this concept, and have not found any good resources on google that explain it in a straightforward manner:

An adjacency matching in an undirected graph G is a collection of disjoint edge pairs in G such that if two edges e and e' are paired, then e and e' share a common endpoint.

I am asking clarification of this as it is concept presented in homework, to which I will have to develop an algorithm to solve for - in other words, I am asking about the question itself and not the answer.

Answer 1

A matching is a labeling such if two edges share a vertex, at most one of them is matched.

So if we have a path $a - b - c - d$, we add edges $\{a, b\}$ and $\{c, d\}$ to the matching. Alternatively, the edge $\{b, c\}$ would solely constitute a matching, as adding any other edge would violate the matching property I stated above.

So visually, we see: $a -*- b -- c -*- d$ as a matching, where $-*-$ represents a matched edge.

Does this concept simply imply that two edges, e->k and j-k, among respective paths say a->b->c->e->k and a->h->i->j->k are adjacency matching?

No, $\{e, k\}$ and $\{j, k\}$ are not both in the matching, as they share a common vertex- k. At most one of them is in the matching.

This doesn't imply that each edge along the path that ultimately shares a common endpoint down the road are included, correct?

This would violate the definition of a matching.

In other words, a maximum adjacency matching would be all adjacency matching pairs in graph G, as specified in the e->k and j-k example?

No, as per the above explanation.