en>Magioladitis: Fix section names per WP:EXTLINKS + general fixes using AWB (8545)

2012-11-12T18:08:16Z

Fix section names per WP:EXTLINKS + general fixes using AWB (8545)

← Older revision		Revision as of 20:08, 12 November 2012
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	~~Claude is her~~ [~~http~~:~~//www~~.~~caribbeansocialnetwork~~.~~com/groups/easy~~-~~diy~~-~~repair-strategies~~-for~~-your-automobile/ extended car warranty~~] ~~name~~ and ~~she completely digs that name~~. ~~Bookkeeping is how he~~ [~~http:~~//~~phoenixbiofuel~~.~~com~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/60/~~userId~~/~~184800~~/~~Default.aspx extended car warranty] car extended auto warranty warranty~~ [~~http:~~/~~/Automobiles.Honda.com/certified~~-~~pre~~-~~owned/warranty~~.~~aspx supports~~] ~~his family~~ and ~~his salary has been really fulfilling~~. ~~The [http~~://~~www~~.~~volvocars.com~~/~~us/sales-services/sales/volvo_certified_preowned/Pages/default.aspx favorite pastime~~] for ~~my [http://Www~~.~~Popularmechanics~~.~~com/cars~~/~~how~~-~~to/repair/how~~-to-~~get~~-a-~~used~~-~~car-warranty~~-and-~~not~~-~~get~~-~~screwed-6654348 children~~] and ~~me is playing crochet and now I'm trying to make money with it. Kansas is our beginning location and my mothers and~~ ~~auto warranty fathers live close by~~.~~<br><br>Here is my weblog~~: [~~http~~:~~//beyoufirst.org/ActivityFeed/MyProfile/tabid/62/userId/31300/Default.aspx extended car warranty~~]		[[File:Graph toughness.svg\|thumb\|240px\|In this graph, removing the four red vertices would produce four connected components (depicted in four different colours). However, there is no set of ''k'' vertices whose removal leaves more than ''k'' components. Therefore, its toughness is exactly 1.]]
			In [[graph theory]], '''toughness''' is a measure of the [[Connectivity_(graph_theory)\|connectivity]] of a graph. A graph {{math\|''G''}} is said to be {{math\|''t''}}-tough for a given [[real number]] {{mvar\|t}} if, for every [[integer]] {{math\|''k'' > 1}}, {{math\|''G''}} cannot be split into {{math\|''k''}} different [[Connected component (graph theory)\|connected components]] by the removal of fewer than {{math\|''tk''}} vertices. For instance, a graph is {{math\|1}}-tough if the number of components formed by removing a set of vertices is always at most as large as the number of removed vertices. The toughness of a graph is the maximum {{math\|''t''}} for which it is {{math\|''t''}}-tough; this is a finite number for all graphs except the [[complete graph]]s, which by convention have infinite toughness.

			Graph toughness was first introduced by {{harvs\|first=Václav\|last=Chvátal\|authorlink=Václav Chvátal\|year=1973\|txt}}. Since then there has been extensive work by other mathematicians on toughness; the recent survey by {{harvtxt\|Bauer\|Broersma\|Schmiechel\|2006}} lists 99 theorems and 162 papers on the subject.

			==Examples==
			Removing <math>k</math> vertices from a [[path graph]] may split the remaining graph into <math>k+1</math> connected components. The maximum ratio of components to removed vertices is achieved by removing one vertex (from the middle of the path) and splitting it into two components. Therefore, paths are <math>\frac12</math>-tough. In contrast, removing <math>k</math> vertices from a [[cycle graph]] leaves at most <math>k</math> remaining connected components, so the cycle is <math>1</math>-tough.

			==Connection to vertex connectivity==
			If a graph is <math>t</math>-tough, then one consequence (obtained by setting <math>k=2</math>) is that any set of <math>2t-1</math> nodes can be removed without splitting the graph in two. That is, every <math>t</math>-tough graph is also [[k-vertex-connected graph\|<math>2t</math>-vertex-connected]].

			==Connection to Hamiltonicity==
			{{harvtxt\|Chvátal\|1973}} observed that every [[cycle graph\|cycle]], and therefore every [[Hamiltonian path\|Hamiltonian graph]], is {{math\|1}}-tough; that is, being {{math\|1}}-tough is a [[necessary condition]] for a graph to be Hamiltonian. He conjectured that the connection between toughness and Hamiltonicity goes in both directions: that there exists a threshold <math>t</math> such that every <math>t</math>-tough graph is Hamiltonian. Chvátal's original conjecture that <math>t=2</math> would have proven [[Fleischner's theorem]] but was disproved by {{harvtxt\|Bauer\|Broersma\|Veldman\|2000}} The existence of a larger toughness threshold for Hamiltonicity remains open.

			== References ==
			*{{citation
			\| last1 = Bauer \| first1 = Douglas
			\| last2 = Broersma \| first2 = Hajo
			\| last3 = Schmeichel \| first3 = Edward
			\| doi = 10.1007/s00373-006-0649-0
			\| issue = 1
			\| journal = Graphs and Combinatorics
			\| pages = 1–35
			\| title = Toughness in graphs—a survey
			\| volume = 22
			\| year = 2006
			\| mr = 2221006}}.
			*{{citation
			\| last1 = Bauer \| first1 = D.
			\| last2 = Broersma \| first2 = H. J.
			\| last3 = Veldman \| first3 = H. J.
			\| contribution = Not every 2-tough graph is Hamiltonian
			\| doi = 10.1016/S0166-218X(99)00141-9
			\| edition = 1-3
			\| journal = Discrete Applied Mathematics
			\| mr = 1743840
			\| pages = 317–321
			\| title = Proceedings of the 5th Twente Workshop on Graphs and Combinatorial Optimization (Enschede, 1997)
			\| volume = 99
			\| year = 2000}}.
			*{{citation
			\| last = Chvátal \| first = Václav \| author-link = Václav Chvátal
			\| doi = 10.1016/0012-365X(73)90138-6
			\| issue = 3
			\| journal = [[Discrete Mathematics (journal)\|Discrete Mathematics]]
			\| pages = 215–228
			\| title = Tough graphs and Hamiltonian circuits
			\| volume = 5
			\| year = 1973
			\| mr = 0316301}}.

			[[Category:Graph connectivity]]
			[[Category:Graph invariants]]

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Homologous

2012-08-16T05:50:10Z

Repairing links to disambiguation pages - You can help! - Homologous

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en>Magioladitis: Fix section names per WP:EXTLINKS + general fixes using AWB (8545)

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Homologous