Graph Theory: Lecture No. 5
Gis a factor critical graph if for each
v∈V(G),G−v(the graph obtained by
removing the vertex vfrom G) has a perfect
matching.
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Graph Theory: Perfect Matchings and Factor Critical Graphs, Slides of Design and Analysis of Algorithms
Tutte's theorem in graph theory, which states that a graph g has a perfect matching if and only if there exists a subset s of its vertices such that the number of components in g - s equals the size of s, and every odd component in g - s has at least 3 edges connecting to s. The document also mentions that every bridge-less cubic graph has a perfect matching.
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G is a factor critical graph if for each v ∈ V (G ), G − v (the graph obtained by removing the vertex v from G ) has a perfect matching.
For any graph G , there exists S ⊆ V (G ) such that the following two properties are satisfied:
1 Consider the bipartite graph obtained by contracting each component of G − S and deleting the edges with both end points in S. In this graph, there is a matching of S. 2 The induced subgraph on each component is factor critical.
The second part of the statement follows immediately from the first part.
If such a set exists in G , then that is a bad set, if G does not have a perfect matching. So, Tutte’s theorem follows immediately from the above statement.
For any S, each odd component has at least 3 edges going to S. Thus there are at least q(G − S). 3 edges reaching S. Since the degree of each vertex is only 3 , we get |S|. 3 ≥ q(G − S). 3 and therefore Tutte’s condition is satisfied.