Search results
Results from the WOW.Com Content Network
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 = (V, E), a perfect matching in G is a subset M of edge set E, such that every vertex in the vertex set V is adjacent to exactly one edge in M. A perfect matching is also called a 1-factor; see Graph factorization ...
Matching (graph theory) In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. [1] In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated ...
In graph theory, the blossom algorithm is an algorithm for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, [ 1 ] and published in 1965. [ 2 ] Given a general graph G = (V, E), the algorithm finds a matching M such that each vertex in V is incident with at most one edge in M and |M| is maximized.
Creating a matching Using a shortcut heuristic on the graph created by the matching above. The algorithm of Christofides and Serdyukov follows a similar outline but combines the minimum spanning tree with a solution of another problem, minimum-weight perfect matching. This gives a TSP tour which is at most 1.5 times the optimal.
A fractional matching in a graph is an assignment of non-negative weights to each edge, such that the sum of weights adjacent to each vertex is at most 1. A fractional matching is X-perfect if the sum of weights adjacent to each vertex is exactly 1. The following are equivalent for a bipartite graph G = (X+Y, E): [13] G admits an X-perfect ...
A matching is a special case of a fractional matching in which all fractions are either 0 or 1. The size of a fractional matching is the sum of fractions of all hyperedges. The fractional matching number of a hypergraph H is the largest size of a fractional matching in H. It is often denoted by ν* (H). [3]
The algorithm addresses the problem that T is not a tour by identifying all the odd degree vertices in T; since the sum of degrees in any graph is even (by the Handshaking lemma), there is an even number of such vertices. The algorithm finds a minimum-weight perfect matching M among the odd-degree ones.
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 ...