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294:...has order 6, not 7 right?... =The dual of a "graph" is not always a "graph" given that the Knowledge has chosen the definition of "graph" to be the one that excludes loops and parallel edges (what is being called a "simple graph" in many books) the dual of a graph is not always a graph. Every connected component of a graph has to be 3-edge connected for its dual to be a graph. A cut edge corresponds to a loop in the dual, and an edge cut of two corresponds to parallel edges in the dual.
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1847:"However, this does not work for shortest path trees, even approximately: there exist planar graphs such that, for every pair of a spanning tree in the graph and a complementary spanning tree in the dual graph, at least one of the two trees has distances that are significantly longer than the distances in its graph."
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The article uses the word "complementary", applied to a graph, to mean "corresponding". It also writes once of "the complement of the graph in the manifold", in the topological sense. All of these may confuse readers who are familiar with the concept of "complement graph". I think confusion could be
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I don't know what means "the dual of a graph is not always a graph", even if we restrict the word graph to "simple graph". Although the definition is not so precise, the term "dual graph" means that it is a graph. Of course, some plane graph won't have a dual graph. Also, it is sometimes written "the
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One reference is authored by the primary
Knowledge author (also nominator). I have checked this reference in detail, and confirm that it was published in conference proceedings, has been broadly cited, and is relevant and balanced in this article. This does not constitute bias or original research
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One reference is authored by the primary
Knowledge author (also nominator). I have checked this reference in detail, and confirm that it was published in conference proceedings, has been broadly cited, and is relevant and balanced in this article. This does not constitute bias or original research
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If it uses complementary to mean corresponding, it's a mistake. It should mean "not corresponding". Could you point out where that happens? Barring such mistakes, the word complement is used in two different senses here: (1) the set of edges not belonging to some particular edge set, and (2) the set
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It is important, and mentioned in the lead, but should probably be elsewhere as well. Maybe as another subsection of the new "Variations" section? If you know of good sources for nonplanar duality that would be helpful (I think I know the subject well enough to write it without sources, but that way
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Re the images: I moved the text on
Intercpunetring.png into an Information template and fixed the link to the user, added text attributing the other image (although my own belief is that the graphs themselves are non-copyrightable, so an image that completely redraws them with a different layout as
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What's "this" which is not working? According to the cited source, "A spanning tree T in a finite planar connected graph G determines a dual spanning tree T∗ in the dual graph G∗ such that T and T∗ do not intersect. We show that it is not always possible to find T in G such that the diameters of T
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really confusing. If I did not already know what it ts trying to say, I don't think I'd be able to figure it out. The footnote specifies "Here we consider that graphs may have loops and multiple edges to avoid uncommon considerations.". If that is true, then the statement should read
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What is the dual to this? The 4-6 edge does not "separate two faces from one another", so therefore does not get an edge in the dual? In which case how do you get node 6 of G back from H? Or should the dual have a loop from the outside "face" to
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The article is almost all about plane graphs – and to its creators' credit, it makes this explicit. The embedding of graphs in other surfaces is something that interests me more than most, so please don't attach too much importance to this comment.
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Some graphs can't be embedded in the plane, but all can be embedded in some surface. Even those that can be embedded in the plane can generally be embedded in some other surface. And duality of graphs is surface dependent. Two examples:
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No. "Polyhedral graph" is a technical term referring only to convex polyhedra. A polyhedral graph must be 3-connected (deleting up to two vertices keeps it connected) while in this case deleting the one shared vertex disconnects it.
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It still has two faces on each side of it, they're just the same face as each other, so the dual vertex corresponding to that face has a self-loop. I've updated the article in an attempt to clarify this, as well as to link
1653:"there also exist self-dual graphs that are not polyhedral, such as the one shown". The one shown seems to be two tetrahedra connected by sharing a vertex in common. Is that still a polyhedron, albeit not convex? --
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It is topologically equivalent. There is a one-to-one matching of the points in top-G and bottom-G such that if a point A is connected to a point B in one, then A's equivalent connects to B's equivalent in the
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00:18, 22 March 2007 (UTC) (sorry I forgot to sign) Also, the
Whitney criterion will also be false. It seems that there is a strong evidence that duality should be introduced for multigraphs with loops allowed.
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I added a history section. Applications will require a bit more thought and research. Presumably it should at least include the duality of series-parallel circuits in CMOS design but there should be others.
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In this section it would help to assert that all convex polyhedra (3d) can be represented as a plane graph, to connect with the definition used of a dual graph. I was led astray here because the link to
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this one does has no actual copyright dependence on its predecessor), and forwarded the emails to OTRS. I assume it may take a little while for OTRS to read them and update the image data accordingly. —
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Wouldn't it be better to speak about multigraphs rather than graphs to introduce dual. If we restrict ourselves to simple loopless graphs, the figure for non-isomorphic duals is wrong (parallel edges).
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You can decompose into minimum spanning tree and dual maximum spanning tree, but you can't decompose into shortest path tree and another tree where you have any control over the lengths of the paths. —
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graph can have non-isomorphic dual graphs." However, the graph G is not identical in the top and bottom of the figure (the left-most vertex has been shifted to the interior). This is confusing to me.
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It would be nice if there were some references to the application of dual graph in a discipline of science, such as
Physics, Geology, Biology, Engineering, Computer Science, etc...
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In the article there is nothing about diameters. As shortest path trees are spanning trees, previous statements about spanning trees must be true for shortest path trees as well.
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Should his page take note that the planar projections of the regular polyhedra come in dual pairs (tetrahedron-tetrahedron, hexahedron-octahedron, and dodecahedron-icosahedron)?
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avoided by replacing "complementary" by "corresponding". I would also like to avoid the phrase "complement of the graph", but cannot at present think of a good way to do it.
1616:, which includes higher dimensions. Since I knew that a 4d hypercube graph wasn't a planar graph, it seemed to contradict the idea that you could necessarily find a dual. --
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It would be tough and somewhat heavy on detail for the general reader, but is roughly appropriate for the more mathematically inclined audience who would come to this page.
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Thank you for your response. I was wrong about "complementary", the article uses it correctly throughout. This just shows that I don't read definitions carefully enough.
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We're also using "connected component" in a different sense in the nonplanar section than elsewhere, so it might be a good idea to avoid that one too if possible. —
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Rewrote this section to give more of an introduction to polyhedral graphs, especially given your question in the next section of the review on self-dual graphs. —
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of points not belonging to some particular point set. I don't know of a concise and accurate way to word those two different senses differently from each other. —
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I think, this page should adopt the clear distinction between the geometric dual and the combinatorial dual; just look up any graph theory text book. --
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Perhaps a little history of when the concept of duals was first developed? Maybe some words on the wider applications/uses of this concept?
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it is misleading to say they come in pairs: a graph G can have two duals G' and G" depending on its plane drawing - as explained later.
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Also the dual of the dual is not the original graph if the original is disconnected. (See Bondy+Murty/Graph Theory and
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can't be embedded in the plane, but can be embedded in the genus-3 orientable surface. There, its dual is the 7-regular Klein graph.
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It appears (to my untrained eye) that the figure for non-isomorphhic duals is in error. It is supposed to illustrate how "the
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and T∗ are both within a uniform multiplicative constant (independent of G) of the diameters of their ambient graphs."
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could do with a description template and ensuring that the "author" is linked properly to the (now-renamed) uploader.
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Is it true that a dual graph is always biconnected? If so then this fact should be added to the article. (unsigned)
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The concept requires both multiple edges and loops. That is how it is nearly always done in technical places.
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Spelling and grammar are fine. It's clear and concise enough, and there are wikilinks to required concepts.
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The cube can be embedded in the plane (or equivalently the sphere), where its dual graph is the octahedron, K
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I will continue to try to think of a way to reword the "complement of the graph in the manifold" sentence.
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In the "Spanning trees" section there is a sentence I really don't understand, or it is out of context:
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what about giving the dualism between the
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No it is not true. Take any planar graph G with a cut-vertex, then its dual also has a cut-vertex.
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Related to this, I can't tell how to make a dual when dangling bond's are involved (see pic)...
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Here we consider that graphs may have loops and multiple edges to avoid uncommon considerations.
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I've done some spot tests. Hard to say definitively, but I think plagiarism is unlikely here.
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If I were working on the article, I would mention this. Maybe it's just as well I'm not.
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It may be interesting to present the dual graph of this three-nodes graphs: 1<-: -->
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According to the definition it is not that clear if it must have one or two loops.
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I rewrote much of this section to clarify some of its terms, including this one. —
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And I have no idea what "uncommon considerations" could be.
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Two loops. There is a dual edge for every primal edge. —
1691:"cutset" is used before it is defined and wikilinked. --
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128:. The nomination discussion and review may be seen at
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336:dual" in this article, instead of "a dual graph".
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766:: I see that you have done an excellent job!
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1648:Self duality
1576:
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1431:. media are
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1337:. media are
1334:
1307:
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1208:main aspects
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1123:cited inline
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999:Well-written
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910:Instructions
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660:— Preceding
656:
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269:Mid-priority
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194:Mid‑priority
172:WikiProjects
114:
105:
102:Did you know
101:
91:Did you know
89:
81:
80:A fact from
52:
43:please do so
31:
30:
26:
1341:with their
1304:Illustrated
924:transcluded
700:Klein graph
619:plane graph
604:is a plane
602:plane graph
471:is a graph
434:—Preceding
244:Mathematics
235:mathematics
191:Mathematics
107:dual graphs
96:check views
1944:Categories
1723:Uniqueness
1379:User:Drini
1373:The image
1356:The image
1175:plagiarism
1074:Verifiable
877:Authorship
863:GA toolbox
698:The cubic
606:multigraph
82:Dual graph
37:under the
27:Dual graph
1580:lovely --
986:Attribute
936:Reviewer:
900:Templates
891:Reviewing
856:GA Review
717:but with
140:Knowledge
104:... that
86:Main Page
1433:relevant
1310:such as
1291:edit war
949:contribs
905:Criteria
713:, like K
674:contribs
662:unsigned
475:so that
436:unsigned
409:contribs
397:unsigned
318:contribs
306:unsigned
162:GA-class
47:reassess
1756:Linked
1538:itself?
1263:Neutral
1056:fiction
824:Maproom
791:Maproom
768:Maproom
727:Maproom
627:Maproom
579:Gabriel
420:example
401:Tejas81
310:Taxipom
271:on the
88:in the
1912:MClerc
1345:, and
1339:tagged
1312:images
1287:Stable
1058:, and
1048:layout
517:Abdull
509:as in
374:other.
168:scale.
54:Review
1785:99of9
1744:99of9
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1320:audio
1318:, or
1316:video
1308:media
1077:with
982:Rate
967:99of9
939:99of9
926:from
741:lies
707:2,2,2
543:McKay
487:is a
361:Kmote
348:McKay
119:trees
112:mazes
1931:talk
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1889:talk
1875:talk
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1577:Done
1558:talk
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1508:Lead
1491:talk
1477:talk
1413:Done
1393:Done
1368:Done
1221:Done
1121:are
971:talk
943:talk
842:talk
828:talk
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795:talk
772:talk
751:talk
745:.) —
731:talk
670:talk
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583:talk
563:talk
547:talk
521:talk
515:? --
479:and
459:Let
444:talk
405:talk
380:talk
365:talk
357:same
314:talk
1907:3.
1885:SyP
1856:SyP
1242:).
1173:or
719:two
507:Cut
491:of
489:cut
467:of
454:Cut
338:pom
328:pom
323:pom
263:Mid
49:it.
1946::
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1429:6b
1385:.
1353:.
1335:6a
1322::
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1301:6.
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1260:.
1236:3b
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1186:3.
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1081::
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