Graphs with $operatorname{diam}(G)=2operatorname{rad}(G)$
$begingroup$
When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?
I know path graphs with odd vertices are such graphs, but is there a more general graph?
graph-theory
edited Nov 13 '18 at 10:40
Bernard
124k742117
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
$endgroup$
add a comment |
$begingroup$
When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?
I know path graphs with odd vertices are such graphs, but is there a more general graph?
graph-theory
edited Nov 13 '18 at 10:40
Bernard
124k742117
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
$endgroup$
1
$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00
1
$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38
1
$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30
add a comment |
$begingroup$
When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?
I know path graphs with odd vertices are such graphs, but is there a more general graph?
graph-theory
edited Nov 13 '18 at 10:40
Bernard
124k742117
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
$endgroup$
When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?
I know path graphs with odd vertices are such graphs, but is there a more general graph?
graph-theory
graph-theory
edited Nov 13 '18 at 10:40
Bernard
124k742117
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
edited Nov 13 '18 at 10:40
Bernard
124k742117
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
edited Nov 13 '18 at 10:40
Bernard
124k742117
edited Nov 13 '18 at 10:40
Bernard
124k742117
edited Nov 13 '18 at 10:40
Bernard
124k742117
124k742117
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
asked Nov 13 '18 at 9:58
Ankit KumarAnkit Kumar
1,542221
1,542221
1
$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00
1
$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38
1
$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30
add a comment |
1
$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00
1
$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38
1
$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30
1
1
$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00
$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00
1
1
$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38
$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38
1
1
$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30
$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:
- A tree has this property if and only if it has a center vertex.
- An interval graph has this property if and only if its diameter is even.
The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
$endgroup$
$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996545%2fgraphs-with-operatornamediamg-2-operatornameradg%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:
- A tree has this property if and only if it has a center vertex.
- An interval graph has this property if and only if its diameter is even.
The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
$endgroup$
$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39
add a comment |
$begingroup$
As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:
- A tree has this property if and only if it has a center vertex.
- An interval graph has this property if and only if its diameter is even.
The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
$endgroup$
$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39
add a comment |
$begingroup$
As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:
- A tree has this property if and only if it has a center vertex.
- An interval graph has this property if and only if its diameter is even.
The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
$endgroup$
As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:
- A tree has this property if and only if it has a center vertex.
- An interval graph has this property if and only if its diameter is even.
The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
edited Dec 23 '18 at 6:42
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
answered Dec 23 '18 at 6:37
aleph_twoaleph_two
27412
27412
$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39
add a comment |
$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39
$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39
$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996545%2fgraphs-with-operatornamediamg-2-operatornameradg%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00
1
$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38
1
$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30