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Two famous properties are called augmenting paths and alternating paths, which are used to quickly determine whether a graph contains a maximum, or minimum, matching, or the matching can be further improved.
shows all the edges, in blue, that connect the bipartite graph.
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Graph matching problems are very common in daily activities.
From online matchmaking and dating sites, to medical residency placement programs, matching algorithms are used in areas spanning scheduling, planning, pairing of vertices, and network flows.
Hopcroft-Karp works by repeatedly increasing the size of a partial matching via augmenting paths.
It then constructs a tree using a breadth-first search in order to find an augmenting path.
If the search finds an augmenting path, the matching gains one more edge.
A variety of other graph labeling problems, and respective solutions, exist for specific configurations of graphs and labels; problems such as graceful labeling, harmonious labeling, lucky-labeling, or even the famous graph coloring problem.
A common bipartite graph matching algorithm is the Hungarian maximum matching algorithm, which finds a maximum matching by finding augmenting paths.
Augmenting paths in matching problems are closely related to augmenting paths in maximum flow problems, such as the max-flow min-cut algorithm, as both signal sub-optimality and space for further refinement.