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The half graph has a unique perfect matching. This is straightforward to see by induction: must be matched to its only neighbor, , and the remaining vertices form another half graph. More strongly, every bipartite graph with a unique perfect matching is a subgraph of a half graph. [4]
A matching in a graph is a set of edges no two of which share an endpoint, and a matching is maximum if no other matching has more edges. [2] It is obvious from the definition that any vertex-cover set must be at least as large as any matching set (since for every edge in the matching, at least one vertex is needed in the cover).
A maximum matching (also known as maximum-cardinality matching [2]) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number of a graph G is the size of a maximum matching. Every maximum matching is maximal, but not every maximal matching is a maximum matching.
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.
The set C constructed this way is a vertex cover: suppose that an edge e is not covered by C; then M ∪ {e} is a matching and e ∉ M, which is a contradiction with the assumption that M is maximal. Furthermore, if e = { u , v } ∈ M , then any vertex cover – including an optimal vertex cover – must contain u or v (or both); otherwise the ...
The Coachella Valley Invitational will include nearly half of the teams in Major League Soccer and the National Women's Soccer League as the preseason event continues to get bigger and bigger ...
But matching sets are just as easy to slip into — and on top of that, you can utilize the separates in other ways! Mixing and matching the pieces fr This Matching Set Is Key to Creating So Many ...
Maximum cardinality matching is a fundamental problem in graph theory. [1] We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this ...