A matching graph (
couplage d'un graphe) is a graph $G'$ having a subset of $G$ edges without common vertices (two vertices are not adjacent). To summarize, you simply have to pick paths of 2 vertices, while making sure that you didn't use a vertex in another path you picked. The edges you used in your paths are forming the matching graph.
- Maximal: if we add one more edge of $G$, then this isn't a matching graph anymore
- Maximum: we can't make a matching graph with more edges
- Perfect/Complete: every vertex of $G$ is used in $G'$ in a path.
A perfect matching is both maximum and maximal. A maximum matching is also maximal.
- pick an edge
- delete all edges incident to your two incident vertex (aside from the one you picked)
- mark the edge as "picked"
- again, until all the edges are either "picked" or "removed"
Give a maximal matching, maximum matching, and perfect matching of the graph $G$.
Note: this graph is called the Petersen graph.
- red: picked
- blue: removed
Using the algorithm, I could get the following maximal matching
I found this perfect matching, which means that I also found a maximal and a maximum matching