What does it mean that a graph dual is not unique, because the dual depends on the particular embedding?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












What does it mean that a graph dual is not unique, because the dual depends on the particular embedding?



I don't get what graph embeddings are and how they're related to the uniqueness of the dual.



Particularly, is drawing the dual meant to be done in planar sense? That one considers as if the graph was projected on to a plane? So one is able to cross edges once in only a particular way, whereas in a "space" representation I don't think the embedding would matter, since one'd always draw through the edges the same way.







share|cite|improve this question






















  • I think you calculate a dual (not the dual) by drawing a plane picture of a (planar) graph, then putting a dual vertex in each region. But there are different ways to draw the graph in the plane that lead to different duals. (Posted as a comment not an answer since I haven't worked out details.)
    – Ethan Bolker
    Aug 8 at 18:56










  • @EthanBolker That confuses the notion of the dual a bit though, doesn't it? Since there are many duals, then how do decide, which one to use?
    – mavavilj
    Aug 8 at 18:57










  • You can't decide that in advance. "The dual" is ambinguous. See en.wikipedia.org/wiki/Dual_graph#Uniqueness for a confirmation of my comment. There's an example there.
    – Ethan Bolker
    Aug 8 at 19:01










  • Interpret dual as a function of the embedded graph, or the embedding itself, not as a function of the graph.
    – user582578
    Aug 8 at 19:08















up vote
0
down vote

favorite












What does it mean that a graph dual is not unique, because the dual depends on the particular embedding?



I don't get what graph embeddings are and how they're related to the uniqueness of the dual.



Particularly, is drawing the dual meant to be done in planar sense? That one considers as if the graph was projected on to a plane? So one is able to cross edges once in only a particular way, whereas in a "space" representation I don't think the embedding would matter, since one'd always draw through the edges the same way.







share|cite|improve this question






















  • I think you calculate a dual (not the dual) by drawing a plane picture of a (planar) graph, then putting a dual vertex in each region. But there are different ways to draw the graph in the plane that lead to different duals. (Posted as a comment not an answer since I haven't worked out details.)
    – Ethan Bolker
    Aug 8 at 18:56










  • @EthanBolker That confuses the notion of the dual a bit though, doesn't it? Since there are many duals, then how do decide, which one to use?
    – mavavilj
    Aug 8 at 18:57










  • You can't decide that in advance. "The dual" is ambinguous. See en.wikipedia.org/wiki/Dual_graph#Uniqueness for a confirmation of my comment. There's an example there.
    – Ethan Bolker
    Aug 8 at 19:01










  • Interpret dual as a function of the embedded graph, or the embedding itself, not as a function of the graph.
    – user582578
    Aug 8 at 19:08













up vote
0
down vote

favorite









up vote
0
down vote

favorite











What does it mean that a graph dual is not unique, because the dual depends on the particular embedding?



I don't get what graph embeddings are and how they're related to the uniqueness of the dual.



Particularly, is drawing the dual meant to be done in planar sense? That one considers as if the graph was projected on to a plane? So one is able to cross edges once in only a particular way, whereas in a "space" representation I don't think the embedding would matter, since one'd always draw through the edges the same way.







share|cite|improve this question














What does it mean that a graph dual is not unique, because the dual depends on the particular embedding?



I don't get what graph embeddings are and how they're related to the uniqueness of the dual.



Particularly, is drawing the dual meant to be done in planar sense? That one considers as if the graph was projected on to a plane? So one is able to cross edges once in only a particular way, whereas in a "space" representation I don't think the embedding would matter, since one'd always draw through the edges the same way.









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 8 at 18:56

























asked Aug 8 at 18:52









mavavilj

2,470730




2,470730











  • I think you calculate a dual (not the dual) by drawing a plane picture of a (planar) graph, then putting a dual vertex in each region. But there are different ways to draw the graph in the plane that lead to different duals. (Posted as a comment not an answer since I haven't worked out details.)
    – Ethan Bolker
    Aug 8 at 18:56










  • @EthanBolker That confuses the notion of the dual a bit though, doesn't it? Since there are many duals, then how do decide, which one to use?
    – mavavilj
    Aug 8 at 18:57










  • You can't decide that in advance. "The dual" is ambinguous. See en.wikipedia.org/wiki/Dual_graph#Uniqueness for a confirmation of my comment. There's an example there.
    – Ethan Bolker
    Aug 8 at 19:01










  • Interpret dual as a function of the embedded graph, or the embedding itself, not as a function of the graph.
    – user582578
    Aug 8 at 19:08

















  • I think you calculate a dual (not the dual) by drawing a plane picture of a (planar) graph, then putting a dual vertex in each region. But there are different ways to draw the graph in the plane that lead to different duals. (Posted as a comment not an answer since I haven't worked out details.)
    – Ethan Bolker
    Aug 8 at 18:56










  • @EthanBolker That confuses the notion of the dual a bit though, doesn't it? Since there are many duals, then how do decide, which one to use?
    – mavavilj
    Aug 8 at 18:57










  • You can't decide that in advance. "The dual" is ambinguous. See en.wikipedia.org/wiki/Dual_graph#Uniqueness for a confirmation of my comment. There's an example there.
    – Ethan Bolker
    Aug 8 at 19:01










  • Interpret dual as a function of the embedded graph, or the embedding itself, not as a function of the graph.
    – user582578
    Aug 8 at 19:08
















I think you calculate a dual (not the dual) by drawing a plane picture of a (planar) graph, then putting a dual vertex in each region. But there are different ways to draw the graph in the plane that lead to different duals. (Posted as a comment not an answer since I haven't worked out details.)
– Ethan Bolker
Aug 8 at 18:56




I think you calculate a dual (not the dual) by drawing a plane picture of a (planar) graph, then putting a dual vertex in each region. But there are different ways to draw the graph in the plane that lead to different duals. (Posted as a comment not an answer since I haven't worked out details.)
– Ethan Bolker
Aug 8 at 18:56












@EthanBolker That confuses the notion of the dual a bit though, doesn't it? Since there are many duals, then how do decide, which one to use?
– mavavilj
Aug 8 at 18:57




@EthanBolker That confuses the notion of the dual a bit though, doesn't it? Since there are many duals, then how do decide, which one to use?
– mavavilj
Aug 8 at 18:57












You can't decide that in advance. "The dual" is ambinguous. See en.wikipedia.org/wiki/Dual_graph#Uniqueness for a confirmation of my comment. There's an example there.
– Ethan Bolker
Aug 8 at 19:01




You can't decide that in advance. "The dual" is ambinguous. See en.wikipedia.org/wiki/Dual_graph#Uniqueness for a confirmation of my comment. There's an example there.
– Ethan Bolker
Aug 8 at 19:01












Interpret dual as a function of the embedded graph, or the embedding itself, not as a function of the graph.
– user582578
Aug 8 at 19:08





Interpret dual as a function of the embedded graph, or the embedding itself, not as a function of the graph.
– user582578
Aug 8 at 19:08
















active

oldest

votes











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',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);








 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2876440%2fwhat-does-it-mean-that-a-graph-dual-is-not-unique-because-the-dual-depends-on-t%23new-answer', 'question_page');

);

Post as a guest



































active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes










 

draft saved


draft discarded


























 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2876440%2fwhat-does-it-mean-that-a-graph-dual-is-not-unique-because-the-dual-depends-on-t%23new-answer', 'question_page');

);

Post as a guest













































































這個網誌中的熱門文章

How to combine Bézier curves to a surface?

Mutual Information Always Non-negative

Why am i infinitely getting the same tweet with the Twitter Search API?