Graph theory and trees
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Let T be a tree with n vertices,where n greater than or equal to 3.Show that there is a vertex V in T with d(V) greater than or equal to 2 such that every vertex adjacent to V ,except possibly for one ,has degree 1.
graph-theory
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
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add a comment |
$begingroup$
Let T be a tree with n vertices,where n greater than or equal to 3.Show that there is a vertex V in T with d(V) greater than or equal to 2 such that every vertex adjacent to V ,except possibly for one ,has degree 1.
graph-theory
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
$endgroup$
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36
add a comment |
$begingroup$
Let T be a tree with n vertices,where n greater than or equal to 3.Show that there is a vertex V in T with d(V) greater than or equal to 2 such that every vertex adjacent to V ,except possibly for one ,has degree 1.
graph-theory
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
$endgroup$
Let T be a tree with n vertices,where n greater than or equal to 3.Show that there is a vertex V in T with d(V) greater than or equal to 2 such that every vertex adjacent to V ,except possibly for one ,has degree 1.
graph-theory
graph-theory
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
asked Dec 2 '18 at 10:31
Varghese M KVarghese M K
61
61
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36
add a comment |
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 10:36
add a comment |
1 Answer
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Let $T’$ be a subgraph of $T$ induced by the set of all vertices of $T$ which have degree at least $2$. Since $T$ is connected and $nge 3$, $T’$ is non-empty. Since $T’$ is a subgraph of a tree, it is forest, that is a union of mutually disjoint trees. Choose any of these trees and pick as the required vertex $v$ any its leaf, that is a vertex of degree $1$.
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
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1 Answer
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active
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1 Answer
1
active
oldest
votes
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$begingroup$
Let $T’$ be a subgraph of $T$ induced by the set of all vertices of $T$ which have degree at least $2$. Since $T$ is connected and $nge 3$, $T’$ is non-empty. Since $T’$ is a subgraph of a tree, it is forest, that is a union of mutually disjoint trees. Choose any of these trees and pick as the required vertex $v$ any its leaf, that is a vertex of degree $1$.
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
$endgroup$
add a comment |
$begingroup$
Let $T’$ be a subgraph of $T$ induced by the set of all vertices of $T$ which have degree at least $2$. Since $T$ is connected and $nge 3$, $T’$ is non-empty. Since $T’$ is a subgraph of a tree, it is forest, that is a union of mutually disjoint trees. Choose any of these trees and pick as the required vertex $v$ any its leaf, that is a vertex of degree $1$.
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
$endgroup$
add a comment |
$begingroup$
Let $T’$ be a subgraph of $T$ induced by the set of all vertices of $T$ which have degree at least $2$. Since $T$ is connected and $nge 3$, $T’$ is non-empty. Since $T’$ is a subgraph of a tree, it is forest, that is a union of mutually disjoint trees. Choose any of these trees and pick as the required vertex $v$ any its leaf, that is a vertex of degree $1$.
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
$endgroup$
Let $T’$ be a subgraph of $T$ induced by the set of all vertices of $T$ which have degree at least $2$. Since $T$ is connected and $nge 3$, $T’$ is non-empty. Since $T’$ is a subgraph of a tree, it is forest, that is a union of mutually disjoint trees. Choose any of these trees and pick as the required vertex $v$ any its leaf, that is a vertex of degree $1$.
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
answered Dec 3 '18 at 3:35
Alex RavskyAlex Ravsky
41.5k32282
41.5k32282
add a comment |
add a comment |
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36