maximal matching/minimal edge covering theorem
Consider an arbitrary matching on and an arbitrary edge covering on . We will attempt to construct a one-to-one function .
Consider some edge . At least one of the vertices that joins must be in , because is an edge covering and hence every edge is incident with some vertex in . Call this vertex , and let .
Now we will show that one-to-one. Suppose we have two edges where . By the definition of , and must both be incident with . Since is a matching, however, no more than one edge in can be incident with any given vertex in . Therefore , so is one-to-one.
Hence we now have that .
Let be a graph. Let and be a matching and an edge covering on , respectively. If , then is a maximal matching and is a minimal edge covering.
Suppose is not a maximal matching. Then, by definition, there exists another matching where . But then , which violates the above theorem.
Likewise, suppose is not a minimal edge covering. Then, by definition, there exists another covering where . But then , which violates the above theorem.
|Title||maximal matching/minimal edge covering theorem|
|Date of creation||2013-03-22 12:40:05|
|Last modified on||2013-03-22 12:40:05|
|Last modified by||mps (409)|