Perfect matching and maximum matching
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In a graph where a perfect matching is possible, is that perfect matching also always the maximum matching?
graph-theory
asked Jun 24 '15 at 12:14
fragant
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In a graph where a perfect matching is possible, is that perfect matching also always the maximum matching?
graph-theory
asked Jun 24 '15 at 12:14
fragant
15110
What does it mean to be a maximum matching? What does it mean to be a perfect matching? Do you think you could find a matching with more edges than a perfect matching?
– TravisJ
Jun 24 '15 at 13:11
as the perfect matching covers every node in the graph, then it cannot be extended by another edge. Thus it is maximal and in the case of being perfect, it is maximum.
– M.M
Jun 24 '15 at 13:21
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In a graph where a perfect matching is possible, is that perfect matching also always the maximum matching?
graph-theory
asked Jun 24 '15 at 12:14
fragant
15110
In a graph where a perfect matching is possible, is that perfect matching also always the maximum matching?
graph-theory
graph-theory
asked Jun 24 '15 at 12:14
fragant
15110
asked Jun 24 '15 at 12:14
fragant
15110
asked Jun 24 '15 at 12:14
fragant
15110
asked Jun 24 '15 at 12:14
fragant
15110
asked Jun 24 '15 at 12:14
fragant
15110
15110
What does it mean to be a maximum matching? What does it mean to be a perfect matching? Do you think you could find a matching with more edges than a perfect matching?
– TravisJ
Jun 24 '15 at 13:11
as the perfect matching covers every node in the graph, then it cannot be extended by another edge. Thus it is maximal and in the case of being perfect, it is maximum.
– M.M
Jun 24 '15 at 13:21
add a comment |
What does it mean to be a maximum matching? What does it mean to be a perfect matching? Do you think you could find a matching with more edges than a perfect matching?
– TravisJ
Jun 24 '15 at 13:11
as the perfect matching covers every node in the graph, then it cannot be extended by another edge. Thus it is maximal and in the case of being perfect, it is maximum.
– M.M
Jun 24 '15 at 13:21
What does it mean to be a maximum matching? What does it mean to be a perfect matching? Do you think you could find a matching with more edges than a perfect matching?
– TravisJ
Jun 24 '15 at 13:11
What does it mean to be a maximum matching? What does it mean to be a perfect matching? Do you think you could find a matching with more edges than a perfect matching?
– TravisJ
Jun 24 '15 at 13:11
as the perfect matching covers every node in the graph, then it cannot be extended by another edge. Thus it is maximal and in the case of being perfect, it is maximum.
– M.M
Jun 24 '15 at 13:21
as the perfect matching covers every node in the graph, then it cannot be extended by another edge. Thus it is maximal and in the case of being perfect, it is maximum.
– M.M
Jun 24 '15 at 13:21
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
Indeed a perfect matching is an example of a maximum matching; this follows from the definitions:
A perfect matching is a matching which matches all vertices of the graph.
A maximum matching is a matching that contains the largest possible number of edges.
If we added an edge to a perfect matching it would no longer be a matching.
To be a perfect matching of a graph $G=(V,E)$, it must have $|V|/2$ edges, and thus $|V|$ must be even.
It is not necessarily the case that it is "the maximum matching", as there might be multiple maximum matchings. E.g. the blue edges and the orange edges form two different perfect (and maximum) matchings in the graph below:
edited yesterday
tchappy ha
34319
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
add a comment |
up vote
0
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Let $G = (V, E)$ be an undirected graph which has a perfect matching $Mp$.
Let $Mm$ be a maximum matching of $G$.
Obviously, $2 |Mp| = |V|$.
Obviously, $2 |Mm| leq |V|$.
So, $|Mm| leq frac{|V|}{2} = |Mp|$.
So, $Mp$ is a maximum matching.
answered yesterday
tchappy ha
34319
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Indeed a perfect matching is an example of a maximum matching; this follows from the definitions:
A perfect matching is a matching which matches all vertices of the graph.
A maximum matching is a matching that contains the largest possible number of edges.
If we added an edge to a perfect matching it would no longer be a matching.
To be a perfect matching of a graph $G=(V,E)$, it must have $|V|/2$ edges, and thus $|V|$ must be even.
It is not necessarily the case that it is "the maximum matching", as there might be multiple maximum matchings. E.g. the blue edges and the orange edges form two different perfect (and maximum) matchings in the graph below:
edited yesterday
tchappy ha
34319
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
add a comment |
up vote
1
down vote
accepted
Indeed a perfect matching is an example of a maximum matching; this follows from the definitions:
A perfect matching is a matching which matches all vertices of the graph.
A maximum matching is a matching that contains the largest possible number of edges.
If we added an edge to a perfect matching it would no longer be a matching.
To be a perfect matching of a graph $G=(V,E)$, it must have $|V|/2$ edges, and thus $|V|$ must be even.
It is not necessarily the case that it is "the maximum matching", as there might be multiple maximum matchings. E.g. the blue edges and the orange edges form two different perfect (and maximum) matchings in the graph below:
edited yesterday
tchappy ha
34319
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Indeed a perfect matching is an example of a maximum matching; this follows from the definitions:
A perfect matching is a matching which matches all vertices of the graph.
A maximum matching is a matching that contains the largest possible number of edges.
If we added an edge to a perfect matching it would no longer be a matching.
To be a perfect matching of a graph $G=(V,E)$, it must have $|V|/2$ edges, and thus $|V|$ must be even.
It is not necessarily the case that it is "the maximum matching", as there might be multiple maximum matchings. E.g. the blue edges and the orange edges form two different perfect (and maximum) matchings in the graph below:
edited yesterday
tchappy ha
34319
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
Indeed a perfect matching is an example of a maximum matching; this follows from the definitions:
A perfect matching is a matching which matches all vertices of the graph.
A maximum matching is a matching that contains the largest possible number of edges.
If we added an edge to a perfect matching it would no longer be a matching.
To be a perfect matching of a graph $G=(V,E)$, it must have $|V|/2$ edges, and thus $|V|$ must be even.
It is not necessarily the case that it is "the maximum matching", as there might be multiple maximum matchings. E.g. the blue edges and the orange edges form two different perfect (and maximum) matchings in the graph below:
edited yesterday
tchappy ha
34319
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
edited yesterday
tchappy ha
34319
edited yesterday
tchappy ha
34319
edited yesterday
tchappy ha
34319
34319
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
answered Jun 24 '15 at 23:26
Rebecca J. Stones
20.8k22781
20.8k22781
add a comment |
add a comment |
up vote
0
down vote
Let $G = (V, E)$ be an undirected graph which has a perfect matching $Mp$.
Let $Mm$ be a maximum matching of $G$.
Obviously, $2 |Mp| = |V|$.
Obviously, $2 |Mm| leq |V|$.
So, $|Mm| leq frac{|V|}{2} = |Mp|$.
So, $Mp$ is a maximum matching.
answered yesterday
tchappy ha
34319
add a comment |
up vote
0
down vote
Let $G = (V, E)$ be an undirected graph which has a perfect matching $Mp$.
Let $Mm$ be a maximum matching of $G$.
Obviously, $2 |Mp| = |V|$.
Obviously, $2 |Mm| leq |V|$.
So, $|Mm| leq frac{|V|}{2} = |Mp|$.
So, $Mp$ is a maximum matching.
answered yesterday
tchappy ha
34319
add a comment |
up vote
0
down vote
up vote
0
down vote
Let $G = (V, E)$ be an undirected graph which has a perfect matching $Mp$.
Let $Mm$ be a maximum matching of $G$.
Obviously, $2 |Mp| = |V|$.
Obviously, $2 |Mm| leq |V|$.
So, $|Mm| leq frac{|V|}{2} = |Mp|$.
So, $Mp$ is a maximum matching.
answered yesterday
tchappy ha
34319
Let $G = (V, E)$ be an undirected graph which has a perfect matching $Mp$.
Let $Mm$ be a maximum matching of $G$.
Obviously, $2 |Mp| = |V|$.
Obviously, $2 |Mm| leq |V|$.
So, $|Mm| leq frac{|V|}{2} = |Mp|$.
So, $Mp$ is a maximum matching.
answered yesterday
tchappy ha
34319
answered yesterday
tchappy ha
34319
answered yesterday
tchappy ha
34319
answered yesterday
tchappy ha
34319
34319
add a comment |
add a comment |
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What does it mean to be a maximum matching? What does it mean to be a perfect matching? Do you think you could find a matching with more edges than a perfect matching?
– TravisJ
Jun 24 '15 at 13:11
as the perfect matching covers every node in the graph, then it cannot be extended by another edge. Thus it is maximal and in the case of being perfect, it is maximum.
– M.M
Jun 24 '15 at 13:21