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In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = { 1 , 2 , … , n } {\displaystyle V=\{1,2,\dots ,n\}} then the Tutte matrix is an n -by- n matrix A with entries
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G with edges E and vertices V, a perfect matching in G is a subset M of E, such that every vertex in V is adjacent to exactly one edge in M. The adjacency matrix of a perfect matching is a symmetric permutation matrix.
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 ...
The problem of finding a matching with maximum weight in a weighted graph is called the maximum weight matching problem, and its restriction to bipartite graphs is called the assignment problem. If each vertex can be matched to several vertices at once, then this is a generalized assignment problem .
The matching polynomial of a graph G with n vertices is related to that of its complement by a pair of (equivalent) formulas. One of them is a simple combinatorial identity due to Zaslavsky (1981) . The other is an integral identity due to Godsil (1981) .
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 a cubic graph with a perfect matching, the edges that are not in the perfect matching form a 2-factor. By orienting the 2-factor, the edges of the perfect matching can be extended to paths of length three, say by taking the outward-oriented edges. This shows that every cubic, bridgeless graph decomposes into edge-disjoint paths of length ...
An graph (or a component) with an odd number of vertices cannot have a perfect matching, since there will always be a vertex left alone. The goal is to characterize all graphs that do not have a perfect matching. Start with the most obvious case of a graph without a perfect matching: a graph with an odd number of vertices.