Are there $|M^{*}| - |M|$ augmenting paths with respect to $M$ in $G$ which are vertex disjoint?











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I am reading a book about Graph Algorithms written by Takao Asano.

In the book, the author says the following proposition is true, without a proof.

Is the follwing proposition true or false?

If true, please tell me the proof.




Let $G$ be a graph.

Let $M^{*}$ be a maximum matching in $G$.

Let $M$ be a matching in $G$.



Then, there exist $|M^{*}| - |M|$ augmenting paths with respect to $M$ in $G$ which are vertex disjoint.











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    up vote
    1
    down vote

    favorite












    I am reading a book about Graph Algorithms written by Takao Asano.

    In the book, the author says the following proposition is true, without a proof.

    Is the follwing proposition true or false?

    If true, please tell me the proof.




    Let $G$ be a graph.

    Let $M^{*}$ be a maximum matching in $G$.

    Let $M$ be a matching in $G$.



    Then, there exist $|M^{*}| - |M|$ augmenting paths with respect to $M$ in $G$ which are vertex disjoint.











    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I am reading a book about Graph Algorithms written by Takao Asano.

      In the book, the author says the following proposition is true, without a proof.

      Is the follwing proposition true or false?

      If true, please tell me the proof.




      Let $G$ be a graph.

      Let $M^{*}$ be a maximum matching in $G$.

      Let $M$ be a matching in $G$.



      Then, there exist $|M^{*}| - |M|$ augmenting paths with respect to $M$ in $G$ which are vertex disjoint.











      share|cite|improve this question













      I am reading a book about Graph Algorithms written by Takao Asano.

      In the book, the author says the following proposition is true, without a proof.

      Is the follwing proposition true or false?

      If true, please tell me the proof.




      Let $G$ be a graph.

      Let $M^{*}$ be a maximum matching in $G$.

      Let $M$ be a matching in $G$.



      Then, there exist $|M^{*}| - |M|$ augmenting paths with respect to $M$ in $G$ which are vertex disjoint.








      graph-theory algorithms matching-theory






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      asked 2 days ago









      tchappy ha

      35019




      35019






















          1 Answer
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          up vote
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          down vote



          accepted










          As always, to compare two matchings, just take the symmetric difference $M mathbin{Delta} M^*$. (That is, take all edges in $M$ or $M^*$, but not both.)



          The components of the subgraph spanned by $M mathbin{Delta} M^*$ are all paths or cycles, because all vertices have degree at most $2$. (At most, they are incident to one edge of $M$ and one edge of $M^*$.) The edges along such a path or cycle must alternate between edges of $M$ and edges of $M^*$. So we have the following types of components:




          1. Even cycles, and paths with an even number of edges. We ignore these, because they have the same number of edges from both matchings.


          2. Paths with one more edge from $M$ than from $M^*$. These would be augmenting paths for $M^*$, and so they can't actually exist, because $M^*$ is a maximum matching.


          3. Paths with one more edge from $M^*$ than from $M$. These are augmenting paths for $M$, so they are what we want.



          We know that there must be $k := |M^*| - |M|$ components of the third type, giving us $k$ vertex-disjoint $M$-augmenting paths. We know this because $M mathbin{Delta} M^*$ has $k$ more edges from $M^*$ than from $M$, and only components of the third type can give us more edges from $M^*$ than from $M$ (one extra edge per component).






          share|cite|improve this answer























          • Thank you very much, Misha Lavrov for very clear proof!
            – tchappy ha
            yesterday










          • I think "2. Paths with one more edge from $M$ than from $M^*$." and "3. Paths with one more edge from $M^*$ than from $M$." are correct. I cannot edit that.
            – tchappy ha
            yesterday










          • @tchappyha Thank you for catching that!
            – Misha Lavrov
            yesterday











          Your Answer





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          1 Answer
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          active

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote



          accepted










          As always, to compare two matchings, just take the symmetric difference $M mathbin{Delta} M^*$. (That is, take all edges in $M$ or $M^*$, but not both.)



          The components of the subgraph spanned by $M mathbin{Delta} M^*$ are all paths or cycles, because all vertices have degree at most $2$. (At most, they are incident to one edge of $M$ and one edge of $M^*$.) The edges along such a path or cycle must alternate between edges of $M$ and edges of $M^*$. So we have the following types of components:




          1. Even cycles, and paths with an even number of edges. We ignore these, because they have the same number of edges from both matchings.


          2. Paths with one more edge from $M$ than from $M^*$. These would be augmenting paths for $M^*$, and so they can't actually exist, because $M^*$ is a maximum matching.


          3. Paths with one more edge from $M^*$ than from $M$. These are augmenting paths for $M$, so they are what we want.



          We know that there must be $k := |M^*| - |M|$ components of the third type, giving us $k$ vertex-disjoint $M$-augmenting paths. We know this because $M mathbin{Delta} M^*$ has $k$ more edges from $M^*$ than from $M$, and only components of the third type can give us more edges from $M^*$ than from $M$ (one extra edge per component).






          share|cite|improve this answer























          • Thank you very much, Misha Lavrov for very clear proof!
            – tchappy ha
            yesterday










          • I think "2. Paths with one more edge from $M$ than from $M^*$." and "3. Paths with one more edge from $M^*$ than from $M$." are correct. I cannot edit that.
            – tchappy ha
            yesterday










          • @tchappyha Thank you for catching that!
            – Misha Lavrov
            yesterday















          up vote
          1
          down vote



          accepted










          As always, to compare two matchings, just take the symmetric difference $M mathbin{Delta} M^*$. (That is, take all edges in $M$ or $M^*$, but not both.)



          The components of the subgraph spanned by $M mathbin{Delta} M^*$ are all paths or cycles, because all vertices have degree at most $2$. (At most, they are incident to one edge of $M$ and one edge of $M^*$.) The edges along such a path or cycle must alternate between edges of $M$ and edges of $M^*$. So we have the following types of components:




          1. Even cycles, and paths with an even number of edges. We ignore these, because they have the same number of edges from both matchings.


          2. Paths with one more edge from $M$ than from $M^*$. These would be augmenting paths for $M^*$, and so they can't actually exist, because $M^*$ is a maximum matching.


          3. Paths with one more edge from $M^*$ than from $M$. These are augmenting paths for $M$, so they are what we want.



          We know that there must be $k := |M^*| - |M|$ components of the third type, giving us $k$ vertex-disjoint $M$-augmenting paths. We know this because $M mathbin{Delta} M^*$ has $k$ more edges from $M^*$ than from $M$, and only components of the third type can give us more edges from $M^*$ than from $M$ (one extra edge per component).






          share|cite|improve this answer























          • Thank you very much, Misha Lavrov for very clear proof!
            – tchappy ha
            yesterday










          • I think "2. Paths with one more edge from $M$ than from $M^*$." and "3. Paths with one more edge from $M^*$ than from $M$." are correct. I cannot edit that.
            – tchappy ha
            yesterday










          • @tchappyha Thank you for catching that!
            – Misha Lavrov
            yesterday













          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          As always, to compare two matchings, just take the symmetric difference $M mathbin{Delta} M^*$. (That is, take all edges in $M$ or $M^*$, but not both.)



          The components of the subgraph spanned by $M mathbin{Delta} M^*$ are all paths or cycles, because all vertices have degree at most $2$. (At most, they are incident to one edge of $M$ and one edge of $M^*$.) The edges along such a path or cycle must alternate between edges of $M$ and edges of $M^*$. So we have the following types of components:




          1. Even cycles, and paths with an even number of edges. We ignore these, because they have the same number of edges from both matchings.


          2. Paths with one more edge from $M$ than from $M^*$. These would be augmenting paths for $M^*$, and so they can't actually exist, because $M^*$ is a maximum matching.


          3. Paths with one more edge from $M^*$ than from $M$. These are augmenting paths for $M$, so they are what we want.



          We know that there must be $k := |M^*| - |M|$ components of the third type, giving us $k$ vertex-disjoint $M$-augmenting paths. We know this because $M mathbin{Delta} M^*$ has $k$ more edges from $M^*$ than from $M$, and only components of the third type can give us more edges from $M^*$ than from $M$ (one extra edge per component).






          share|cite|improve this answer














          As always, to compare two matchings, just take the symmetric difference $M mathbin{Delta} M^*$. (That is, take all edges in $M$ or $M^*$, but not both.)



          The components of the subgraph spanned by $M mathbin{Delta} M^*$ are all paths or cycles, because all vertices have degree at most $2$. (At most, they are incident to one edge of $M$ and one edge of $M^*$.) The edges along such a path or cycle must alternate between edges of $M$ and edges of $M^*$. So we have the following types of components:




          1. Even cycles, and paths with an even number of edges. We ignore these, because they have the same number of edges from both matchings.


          2. Paths with one more edge from $M$ than from $M^*$. These would be augmenting paths for $M^*$, and so they can't actually exist, because $M^*$ is a maximum matching.


          3. Paths with one more edge from $M^*$ than from $M$. These are augmenting paths for $M$, so they are what we want.



          We know that there must be $k := |M^*| - |M|$ components of the third type, giving us $k$ vertex-disjoint $M$-augmenting paths. We know this because $M mathbin{Delta} M^*$ has $k$ more edges from $M^*$ than from $M$, and only components of the third type can give us more edges from $M^*$ than from $M$ (one extra edge per component).







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited yesterday

























          answered yesterday









          Misha Lavrov

          41.5k555101




          41.5k555101












          • Thank you very much, Misha Lavrov for very clear proof!
            – tchappy ha
            yesterday










          • I think "2. Paths with one more edge from $M$ than from $M^*$." and "3. Paths with one more edge from $M^*$ than from $M$." are correct. I cannot edit that.
            – tchappy ha
            yesterday










          • @tchappyha Thank you for catching that!
            – Misha Lavrov
            yesterday


















          • Thank you very much, Misha Lavrov for very clear proof!
            – tchappy ha
            yesterday










          • I think "2. Paths with one more edge from $M$ than from $M^*$." and "3. Paths with one more edge from $M^*$ than from $M$." are correct. I cannot edit that.
            – tchappy ha
            yesterday










          • @tchappyha Thank you for catching that!
            – Misha Lavrov
            yesterday
















          Thank you very much, Misha Lavrov for very clear proof!
          – tchappy ha
          yesterday




          Thank you very much, Misha Lavrov for very clear proof!
          – tchappy ha
          yesterday












          I think "2. Paths with one more edge from $M$ than from $M^*$." and "3. Paths with one more edge from $M^*$ than from $M$." are correct. I cannot edit that.
          – tchappy ha
          yesterday




          I think "2. Paths with one more edge from $M$ than from $M^*$." and "3. Paths with one more edge from $M^*$ than from $M$." are correct. I cannot edit that.
          – tchappy ha
          yesterday












          @tchappyha Thank you for catching that!
          – Misha Lavrov
          yesterday




          @tchappyha Thank you for catching that!
          – Misha Lavrov
          yesterday


















           

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