There are at least 2 vertex-disjoint paths between every pair of vertices?
$begingroup$
$G$ is a graph on $n$ vertices and $2n−2$ edges$.$ The edges of G can
be partitioned into two edge-disjoint spanning trees. Which of the
following is NOT true for $G?$
- For every subset of $k$ vertices, the induced subgraph has at most $2k−2$ edges.
- The minimum cut in $G$ has at least $2$ edges.
- There are at least $2$ edge-disjoint paths between every pair of vertices.
- There are at least $2$ vertex-disjoint paths between every pair of vertices.
My explained as $:$
Counter for option $4$ is as follows.
Take two copies of $K_4$$($complete graph on 4 vertices$),$$G_1$ and $G_2$.
Let $V(G_1)={1,2,3,4}$ and $V(G_2)={5,6,7,8}$.
Construct a new graph $G_3$ by using these two graphs $G_1$ and $G_2$ by merging at a vertex, say merge $(4,5)$.
The resultant graph is $2$ edge connected, and of minimum degree $2$ but there exist a cut vertex, the merged vertex.
discrete-mathematics graph-theory trees
edited Dec 28 '18 at 7:10
Old Pro
304214
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
$endgroup$
add a comment |
$begingroup$
$G$ is a graph on $n$ vertices and $2n−2$ edges$.$ The edges of G can
be partitioned into two edge-disjoint spanning trees. Which of the
following is NOT true for $G?$
- For every subset of $k$ vertices, the induced subgraph has at most $2k−2$ edges.
- The minimum cut in $G$ has at least $2$ edges.
- There are at least $2$ edge-disjoint paths between every pair of vertices.
- There are at least $2$ vertex-disjoint paths between every pair of vertices.
My explained as $:$
Counter for option $4$ is as follows.
Take two copies of $K_4$$($complete graph on 4 vertices$),$$G_1$ and $G_2$.
Let $V(G_1)={1,2,3,4}$ and $V(G_2)={5,6,7,8}$.
Construct a new graph $G_3$ by using these two graphs $G_1$ and $G_2$ by merging at a vertex, say merge $(4,5)$.
The resultant graph is $2$ edge connected, and of minimum degree $2$ but there exist a cut vertex, the merged vertex.
discrete-mathematics graph-theory trees
edited Dec 28 '18 at 7:10
Old Pro
304214
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
$endgroup$
$begingroup$
Please be careful with the tags on questions. This question should not be tagged as computer science. Perhaps you saw it in a CS class? That is not the meaning of the tag - only questions that are directly about computer science should have that tag. This question is purely about mathematics.
$endgroup$
– Carl Mummert
Jan 30 '16 at 11:21
add a comment |
$begingroup$
$G$ is a graph on $n$ vertices and $2n−2$ edges$.$ The edges of G can
be partitioned into two edge-disjoint spanning trees. Which of the
following is NOT true for $G?$
- For every subset of $k$ vertices, the induced subgraph has at most $2k−2$ edges.
- The minimum cut in $G$ has at least $2$ edges.
- There are at least $2$ edge-disjoint paths between every pair of vertices.
- There are at least $2$ vertex-disjoint paths between every pair of vertices.
My explained as $:$
Counter for option $4$ is as follows.
Take two copies of $K_4$$($complete graph on 4 vertices$),$$G_1$ and $G_2$.
Let $V(G_1)={1,2,3,4}$ and $V(G_2)={5,6,7,8}$.
Construct a new graph $G_3$ by using these two graphs $G_1$ and $G_2$ by merging at a vertex, say merge $(4,5)$.
The resultant graph is $2$ edge connected, and of minimum degree $2$ but there exist a cut vertex, the merged vertex.
discrete-mathematics graph-theory trees
edited Dec 28 '18 at 7:10
Old Pro
304214
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
$endgroup$
$G$ is a graph on $n$ vertices and $2n−2$ edges$.$ The edges of G can
be partitioned into two edge-disjoint spanning trees. Which of the
following is NOT true for $G?$
- For every subset of $k$ vertices, the induced subgraph has at most $2k−2$ edges.
- The minimum cut in $G$ has at least $2$ edges.
- There are at least $2$ edge-disjoint paths between every pair of vertices.
- There are at least $2$ vertex-disjoint paths between every pair of vertices.
My explained as $:$
Counter for option $4$ is as follows.
Take two copies of $K_4$$($complete graph on 4 vertices$),$$G_1$ and $G_2$.
Let $V(G_1)={1,2,3,4}$ and $V(G_2)={5,6,7,8}$.
Construct a new graph $G_3$ by using these two graphs $G_1$ and $G_2$ by merging at a vertex, say merge $(4,5)$.
The resultant graph is $2$ edge connected, and of minimum degree $2$ but there exist a cut vertex, the merged vertex.
discrete-mathematics graph-theory trees
discrete-mathematics graph-theory trees
edited Dec 28 '18 at 7:10
Old Pro
304214
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
edited Dec 28 '18 at 7:10
Old Pro
304214
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
edited Dec 28 '18 at 7:10
Old Pro
304214
edited Dec 28 '18 at 7:10
Old Pro
304214
edited Dec 28 '18 at 7:10
Old Pro
304214
304214
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
asked Oct 17 '15 at 13:02
Mithlesh UpadhyayMithlesh Upadhyay
2,93782969
2,93782969
$begingroup$
Please be careful with the tags on questions. This question should not be tagged as computer science. Perhaps you saw it in a CS class? That is not the meaning of the tag - only questions that are directly about computer science should have that tag. This question is purely about mathematics.
$endgroup$
– Carl Mummert
Jan 30 '16 at 11:21
add a comment |
$begingroup$
Please be careful with the tags on questions. This question should not be tagged as computer science. Perhaps you saw it in a CS class? That is not the meaning of the tag - only questions that are directly about computer science should have that tag. This question is purely about mathematics.
$endgroup$
– Carl Mummert
Jan 30 '16 at 11:21
$begingroup$
Please be careful with the tags on questions. This question should not be tagged as computer science. Perhaps you saw it in a CS class? That is not the meaning of the tag - only questions that are directly about computer science should have that tag. This question is purely about mathematics.
$endgroup$
– Carl Mummert
Jan 30 '16 at 11:21
$begingroup$
Please be careful with the tags on questions. This question should not be tagged as computer science. Perhaps you saw it in a CS class? That is not the meaning of the tag - only questions that are directly about computer science should have that tag. This question is purely about mathematics.
$endgroup$
– Carl Mummert
Jan 30 '16 at 11:21
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your construction is indeed a counter-example.
Though I don't see why you argue that by talking about edge-connectivity and minimum degree 2 (which is not the case). What you need to do is show that $G_3$ has $2n - 2$ edges, has $2$ edge disjoint spanning trees, but has a cut vertex.
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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%2f1484385%2fthere-are-at-least-2-vertex-disjoint-paths-between-every-pair-of-vertices%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
$begingroup$
Your construction is indeed a counter-example.
Though I don't see why you argue that by talking about edge-connectivity and minimum degree 2 (which is not the case). What you need to do is show that $G_3$ has $2n - 2$ edges, has $2$ edge disjoint spanning trees, but has a cut vertex.
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
$endgroup$
add a comment |
$begingroup$
Your construction is indeed a counter-example.
Though I don't see why you argue that by talking about edge-connectivity and minimum degree 2 (which is not the case). What you need to do is show that $G_3$ has $2n - 2$ edges, has $2$ edge disjoint spanning trees, but has a cut vertex.
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
$endgroup$
add a comment |
$begingroup$
Your construction is indeed a counter-example.
Though I don't see why you argue that by talking about edge-connectivity and minimum degree 2 (which is not the case). What you need to do is show that $G_3$ has $2n - 2$ edges, has $2$ edge disjoint spanning trees, but has a cut vertex.
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
$endgroup$
Your construction is indeed a counter-example.
Though I don't see why you argue that by talking about edge-connectivity and minimum degree 2 (which is not the case). What you need to do is show that $G_3$ has $2n - 2$ edges, has $2$ edge disjoint spanning trees, but has a cut vertex.
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
answered Oct 17 '15 at 14:48
Manuel LafondManuel Lafond
2,612716
2,612716
add a comment |
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%2f1484385%2fthere-are-at-least-2-vertex-disjoint-paths-between-every-pair-of-vertices%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
$begingroup$
Please be careful with the tags on questions. This question should not be tagged as computer science. Perhaps you saw it in a CS class? That is not the meaning of the tag - only questions that are directly about computer science should have that tag. This question is purely about mathematics.
$endgroup$
– Carl Mummert
Jan 30 '16 at 11:21