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| [[File:Halin graph.svg|thumb|A Halin graph.]]
| | By investing in a premium Word - Press theme, you're investing in the future of your website. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. This plugin allows a blogger get more Facebook fans on the related fan page. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. <br><br>It is very easy to install Word - Press blog or website. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. User friendly features and flexibility that Word - Press has to offer is second to none. If you adored this write-up and you would certainly like to get more facts regarding [http://GET7.pw/backup_plugin_439902 wordpress backup] kindly check out the internet site. There are plenty of tables that are attached to this particular database. <br><br>There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. One of the great features of Wordpress is its ability to integrate SEO into your site. Contact Infertility Clinic Providing One stop Fertility Solutions at:. Make sure you have the latest versions of all your plugins are updated. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Automated deal feed integration option to populate your blog with relevant deals. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. Page speed is an important factor in ranking, especially with Google. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| In [[graph theory]], a '''Halin graph''' is a type of [[planar graph]]. It is constructed from a [[tree (graph theory)|tree]] that has at least four vertices, none of which have exactly two neighbors. The tree is drawn in the [[Euclidean plane|plane]] so none of its edges cross; then edges are added that connect all its [[Tree_(graph_theory)#Definitions|leaves]] into a [[cycle (graph theory)|cycle]].<ref name=ency>''Encyclopaedia of Mathematics'', first Supplementary volume, 1988, ISBN 0-7923-4709-9, p. 281, article [http://books.google.com/books?id=3ndQH4mTzWQC&pg=PA281&dq=%22halin+graph%22+-wikipedia&ei=RgLsSKP2A5DetAPHydj2Bg&sig=ACfU3U3IK1TmaSTLW3yoIHaMUJvE3rKFIQ "Halin Graph"], and references therein.</ref> Halin graphs are named after German mathematician [[Rudolf Halin]], who studied them in 1971,<ref name="h71">{{citation
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| | last = Halin | first = R. | authorlink = Rudolf Halin
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| | contribution = Studies on minimally ''n''-connected graphs
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| | location = London
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| | mr = 0278980
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| | pages = 129–136
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| | publisher = Academic Press
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| | title = Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969)
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| | year = 1971}}.</ref> but the [[cubic graph|cubic]] Halin graphs had already been studied over a century earlier by [[Thomas Kirkman|Kirkman]].<ref name="kirkman"/>
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| == Construction ==
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| [[File:Triangular prism as halin graph.svg|thumb|A triangular prism, constructed as a Halin graph from a six-vertex tree]]
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| A Halin graph is constructed as follows. Let <math> T </math> be a tree with more than three vertices, such that no vertex of <math> T </math> has [[degree (graph theory)|degree]] two (that is, no vertex has exactly two neighbors), [[graph embedding|embedded]] in the plane. Then a Halin graph is constructed by adding to <math>T</math> a cycle through each of its leaves, such that the augmented graph remains planar.
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| ==Examples==
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| [[File:Wheel graphs.svg|thumb|[[Wheel graph]]s]]
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| A [[star (graph theory)|star]] is a tree with exactly one internal vertex. Applying the Halin graph construction to a star produces a [[wheel graph]], the graph of a [[pyramid (geometry)|pyramid]]. The graph of a [[triangular prism]] is also a Halin graph: it can be drawn so that one of its rectangular faces is the exterior cycle, and the remaining edges form a tree with four leaves, two interior vertices, and five edges.
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| The [[Frucht graph]], one of the two smallest [[cubic graph]]s with no nontrivial [[graph automorphism]]s, is also a Halin graph.
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| ==Properties==
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| Every Halin graph is [[graph connectivity|3-connected]], meaning that it is not possible to delete two vertices from it and disconnect the remaining vertices. It is edge-minimal 3-connected, meaning that if any one of its edges is removed, the remaining graph will no longer be 3-connected.<ref name=ency/> By [[Steinitz's theorem]], as a 3-connected planar graph, it can be represented as the set of vertices and edges of a [[convex polyhedron]]; that is, it is a [[polyhedral graph]]. And, as with every polyhedral graph, its planar embedding is unique up to the choice of which of its faces is to be the outer face.<ref name=ency/>
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| Every Halin graph is a [[Hamiltonian graph]], and every edge of the graph belongs to a Hamiltonian cycle. Moreover, any Halin graph remains Hamiltonian after deletion of any vertex.<ref name="CNP83"/>
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| Because every tree without vertices of degree 2 contains two leaves that share the same parent, every Halin graph contains a triangle. In particular, it is not possible for a Halin graph to be a [[triangle-free graph]] nor a [[bipartite graph]].
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| More strongly, every Halin graph is almost [[pancyclic graph|pancyclic]], in the sense that it has cycles of all lengths from 3 to ''n'' with the possible exception of a single even length. Moreover, any Halin graph remains almost pancyclic if a single edge is contracted, and every Halin graph without interior vertices of degree three is pancyclic.<ref>{{citation
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| | last = Skowrońska | first = Mirosława
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| | contribution = The pancyclicity of Halin graphs and their exterior contractions
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| | editor1-last = Alspach | editor1-first = Brian R.
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| | editor2-last = Godsil | editor2-first = Christopher D. | editor2-link = Chris Godsil | |
| | pages = 179–194
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| | publisher = Elsevier Science Publishers B.V.
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| | series = Annals of Discrete Mathematics
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| | title = Cycles in Graphs
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| | volume = 27
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| | year = 1985}}.</ref>
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| Every Halin graph has [[treewidth]] at most three.<ref name="bod1">{{citation|title=Planar graphs with bounded treewidth|first=Hans|last=Bodlaender|authorlink=Hans L. Bodlaender|series=Technical Report RUU-CS-88-14|publisher=Department of Computer Science, [[Utrecht University]]|year=1988|url=http://archive.cs.uu.nl/pub/RUU/CS/techreps/CS-1988/1988-14.pdf}}.</ref> Therefore, many graph optimization problems that are [[NP-complete]] for arbitrary planar graphs, such as finding a [[maximum independent set]], may be solved in [[linear time]] on Halin graphs using [[dynamic programming]].<ref name="bod2">{{citation|first=Hans|last=Bodlaender|authorlink=Hans L. Bodlaender|contribution=Dynamic programming on graphs with bounded treewidth|title=Proceedings of the 15th International Colloquium on Automata, Languages and Programming|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|volume=317|pages=105–118|year=1988}}.</ref>
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| The [[dual graph|weak dual]] of an embedded planar graph has vertices corresponding to bounded faces of the planar graph, and edges corresponding to adjacent faces. The weak dual of a Halin graph is always [[biconnected graph|biconnected]] and [[outerplanar graph|outerplanar]]. This property may be used to characterize the Halin graphs: an embedded planar graph is a Halin graph, with the leaf cycle of the Halin graph as the outer face of the embedding, if and only if its weak dual is biconnected and outerplanar.<ref name="sp83">{{citation
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| | last1 = Sysło | first1 = Maciej M.
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| | last2 = Proskurowski | first2 = Andrzej
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| | contribution = On Halin graphs
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| | doi = 10.1007/BFb0071635
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| | pages = 248–256
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Mathematics
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| | title = Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–13, 1981
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| | volume = 1018
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| | year = 1983}}.</ref>
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| ==History==
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| In 1971, Halin introduced the Halin graphs as a class of [[minimal element|minimally]] 3-vertex-connected graphs: for every edge in the graph, the removal of that edge reduces the connectivity of the graph.<ref name="h71"/> These graphs gained in significance with the discovery that many algorithmic problems that were computationally infeasible for arbitrary planar graphs could be solved efficiently on them,<ref name="CNP83"/><ref name="sp83"/> a fact that was later explained to be a consequence of their low treewidth.<ref name="bod1"/><ref name="bod2"/>
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| Prior to Halin's work on these graphs, [[graph enumeration]] problems concerning the [[cubic graph|cubic]] Halin graphs were studied in 1856 by [[Thomas Kirkman]]<ref name="kirkman">{{citation|first=Th. P.|last=Kirkman|authorlink=Thomas Kirkman|title=On the enumeration of ''x''-edra having triedral summits and an (''x'' − 1)-gonal base|journal=Philosophical Transactions of the Royal Society of London|year=1856|pages=399–411|jstor=108592}}.</ref> and in 1965 by [[Hans Rademacher]].<ref>{{citation
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| | last = Rademacher | first = Hans | authorlink = Hans Rademacher
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| | journal = Illinois Journal of Mathematics
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| | mr = 0179682
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| | pages = 361–380
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| | title = On the number of certain types of polyhedra
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| | url = http://projecteuclid.org/euclid.ijm/1256068140
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| | volume = 9
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| | year = 1965}}.</ref> Rademacher calls these graphs '''based polyhedra'''. He defines them as the cubic [[polyhedral graph]]s with ''f'' faces in which one of the faces has ''f'' − 1 sides. The graphs that fit this definition are exactly the cubic Halin graphs.
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| The Halin graphs are sometimes also called '''roofless polyhedra''',<ref name="CNP83">{{citation|title=Halin graphs and the travelling salesman problem|first1=G.|last1=Cornuéjols|first2=D.|last2=Naddef|first3=W. R.|last3=Pulleyblank|journal=Mathematical Programming|volume=26|issue=3|year=1983|pages=287–294|doi=10.1007/BF02591867}}.</ref> but, like "based polyhedra", this name may also refer to the cubic Halin graphs.<ref>{{citation
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| | last1 = Lovász | first1 = L. | author1-link = László Lovász
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| | last2 = Plummer | first2 = M. D. | author2-link = Michael D. Plummer
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| | contribution = On a family of planar bicritical graphs
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| | location = London
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| | mr = 0351915
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| | pages = 103–107. London Math. Soc. Lecture Note Ser., No. 13
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| | publisher = Cambridge Univ. Press
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| | title = Combinatorics (Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth, 1973)
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| | year = 1974}}.</ref> The [[convex polyhedra]] whose graphs are Halin graphs have also been called '''domes'''.<ref>{{citation|contribution=Zipper unfolding of domes and prismoids|first1=Erik D.|last1=Demaine|author1-link=Erik Demaine|first2=Martin L.|last2=Demaine|author2-link=Martin Demaine|first3=Ryuhei|last3=Uehara|title=Proceedings of the 25th Canadian Conference on Computational Geometry (CCCG 2013), Waterloo, Ontario, Canada, August 8–10, 2013|pages=43–48|year=2013|url=http://erikdemaine.org/papers/ZipperDomes_CCCG2013/}}.</ref>
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_198.html Halin graphs], Information System on Graph Class Inclusions.
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| *{{mathworld|title=Halin Graph|id=HalinGraph}}
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| [[Category:Graph families]]
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| [[Category:Planar graphs]]
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