Matching in General Graphs

Posted: August 27th, 2021

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Matching in General Graphs

            Discussions on a perfect matching in graphs should focus centrally on general spanning subgraphs. Despite a high similarity between a 1-factor and perfect matching, there is a precise distinction in that a 1-factor is a spanning 1-regular subgraph of a general graph, while a perfect matching implies a set of edges in the same subgraph. In this case, Tutte’s theory has established that 1-factor graphs must have a sufficient and necessary condition that a set of the graph should feature in the vertices of the same graph as its subsets. Most importantly, the vertex of such a subgraph must match the exterior components of a general graph. Thus, the odd features of a general graph and set need to have their vertices matched so that there is a clear differentiation of the vertices of its sets. Hence, it is evident that this theory holds since the addition of an edge enjoining the two characteristics of a graph and set implies a non-increment of the number of the odd components.

            The discussions on the f-factor of a graph imply that the availability of a spanning subgraph of a general graph coupled with both the 1-factors and 2-factors. Attempting to define each vertex’s degree implies that the sufficiency of Tutte’s condition that f-factor is always a subgraph of a set. The condition proves the point that a graph G has an f-factor if and only if a second graph, H, is constructed from the same graph and f-factor as that of graph G. Notably, the sufficiency of this statement is proved by matching vertices with their subsequent edges in the same f-factor graph. Thus, the algorithm’s application is for testing the sufficiency of 1-factor, which matches them perfectly with the maximum matching in the f-factor graph.