Matching structure graph edges from "summary" of Introduction to Graph Theory by Douglas Brent West
A matching in a graph G is a set of pairwise nonadjacent edges. That is, no two edges in a matching share a common endpoint. The simplest example of a matching is the empty set, which contains no edges. A matching of size 1 consists of a single edge, a matching of size 2 is a pair of nonadjacent edges, and so on. In general, a matching in a graph G is a set of edges no two of which are adjacent. A matching of maximum size is called a maximum matching. If G has a matching of size k, where k is the largest possible, then G has a maximum matching. The size of a maximum matching is denoted by α'(G). The matching k in a graph G is a maximum matching if no other matching in G has more edges than k. A matching of maximum size that saturates every vertex in G is called a perfect matching. If G has a perfect matching, then G is said to be a factor-critical graph. A graph in which every vertex has been saturated by a perfect matching is called a factor-critical graph. If G and H are graphs with vertex set V, then a bijective function f:V→V is called a graph isomorphism if uv is an edge in G if and only if f(u)f(v) is an edge in H. In other words, a graph isomorphism preserves edges. Two graphs G and H are said to be isomorphic, denoted G ≅ H, if there exists a graph isomorphism from G to H. In this case, G and H are structurally identical, differing only in the labels attached to the vertices. Given two graphs G and H, we can consider the problem of determining if there exists a matching in G that is isomorphic to a given matching in H. This problem leads to the concept of matching structure graph edges, which provides a way to compare the structural properties of two graphs based on their matching configurations.
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