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The case of exact graph matching is known as the graph isomorphism problem. [1] The problem of exact matching of a graph to a part of another graph is called subgraph isomorphism problem. Inexact graph matching refers to matching problems when exact matching is impossible, e.g., when the number of vertices in the two graphs are different. In ...
A maximal matching is a matching M of a graph G that is not a subset of any other matching. A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The following figure shows examples of maximal matchings (red) in three graphs. A maximum matching (also known as maximum-cardinality ...
Equivalently, it concerns the density of a different class of graphs, the locally linear graphs in which the neighborhood of every vertex is an induced matching. [4] Neither of these types of graph can have a quadratic number of edges, but constructions are known for graphs of this type with nearly-quadratic numbers of edges. [5]
The fifth corner (1/2,1/2,1/2) does not represent a matching - it represents a fractional matching in which each edge is "half in, half out". Note that this is the largest fractional matching in this graph - its weight is 3/2, in contrast to the three integral matchings whose size is only 1. As another example, in the 4-cycle there are 4 edges.
In this case, the dual graph is cubic and bridgeless, so by Petersen's theorem it has a matching, which corresponds in the original graph to a pairing of adjacent triangle faces. Each pair of triangles gives a path of length three that includes the edge connecting the triangles together with two of the four remaining triangle edges. [3]
The graphs with this property are called locally linear graphs [3] or locally matching graphs. [ 4 ] What is the maximum possible number of edges in a bipartite graph with n {\displaystyle n} vertices on each side of its bipartition, whose edges can be partitioned into n {\displaystyle n} induced subgraphs that are each matchings ?
Exact algorithms for computing the graph edit distance between a pair of graphs typically transform the problem into one of finding the minimum cost edit path between the two graphs. The computation of the optimal edit path is cast as a pathfinding search or shortest path problem , often implemented as an A* search algorithm .
It was prominently criticized in economics by Robert LaLonde (1986), [7] who compared estimates of treatment effects from an experiment to comparable estimates produced with matching methods and showed that matching methods are biased. Rajeev Dehejia and Sadek Wahba (1999) reevaluated LaLonde's critique and showed that matching is a good ...