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| | During the Neolithic Revolution, humans started consuming large amounts of dairy products, beans, cereals, alcohol and salt. This is an important point to be notices that the paleo diet menu was so restrictive and was without any taste as it had a less variety in it. Trail mix ideal snack for breakfast, lunch or dinner. There is no reason to measure portions since the the amount of protein in the diet fills you up and satisfies your appetite, unlike processed foods with their empty carbs and sugar which just make you hungrier and thirstier making it easy to eat too much. During this little bit of facts, we might end up being concentrating on [http://gritsandgroceries.info/sitemap/ Paleo diet plan], this is a common diet regime technique pertaining to reducing body fat, increasing muscle tissues, stopping center conditions, and also increasing cholestrerol levels. <br><br>Sources of potassium are beans, legumes, nuts, dairy products, fruits and vegetables. In order to do well in all ten of the fitness domains, your body must have real food to burn. There are certain instances when one particular desires to create some meals suing a swift recipe with fewer components then the paleo diet program menu recipes are the ideal option so we can't say that we are able to under no circumstances think about these recipes as they may be beneficial at occasions. Advocates of the Paleo diet plan motives that historical gentlemen are considerably more more healthy than contemporary adult males predominantly due to what they take in. Evolution has not caught up with our present day millisecond mentality. <br><br>Store in refrigerator in opaque air-tight container. Almonds can help to reduce cholesterol, lessen the likelihood of cardiac arrest and are also acknowledged with reducing the risk of several malignancies such as breast malignancies. However many others take into consideration Paleo Diet Eating Plan is pretty demanding, it has 1000's of followers. The human body needs a certain amount and quality of fat for certain processes. Listed here are a few ways to help bring you on the right path:. <br><br>Enjoy Paleo Chicken Stir Fry Recipe and Paleo Beef Spinach Salad Recipes. Not an expensive weight loss plan, no surgery and will not starve. It is the indigestible component in certain foods such as most fruits and vegetables which work to clean our digestive system so that we are able to allow only healthy substances into our bodies through our bloodstream. To lose that stubborn belly fat though, you've got to eliminate dairy and grain based products from your diet. He ate fruits and vegetables, which grew in abundance. <br><br>Sunflower seeds, pumpkin seeds and flax seeds all offer healthy proteins. The sad part is, most of us don't even know that we're eating that much. It's a lesson to those who think that going gluten-free automatically means weight loss. Day one - cabbage soup and all sorts of fruits except banana. Although there is no local 'Paleo Club' location for you to go to, many websites have been built to provide support and a community for people choosing to eat Paleo. <br><br>The Paleo diet reduces body fat, intensifies the immune system and increase your output of energy. Here are some items that you will surely love about this. A weakened jawbone can cause teeth to loosen, leaving you at a greater risk for gum disease. Would you comprehend the information as well as facts regarding the foodstuff you make. Some of the most devoted Paleolithic adherents also adopt a workout regimen that is thought to be based more closely on the workouts of our ancestors. |
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| In [[mathematics]] and [[computer science]], '''connectivity''' is one of the basic concepts of [[graph theory]]: it asks for the minimum number of elements (nodes or edges) which need to be removed to disconnect the remaining nodes from each other.<ref name="diestel"/> It is closely related to the theory of [[flow network|network flow]] problems. The connectivity of a graph is an important measure of its robustness as a network.
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| ==Definitions of components, cuts and connectivity==
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| In an [[undirected graph]] ''G'', two [[vertex (graph theory)|vertices]] ''u'' and ''v'' are called connected if ''G'' contains a [[Path (graph theory)|path]] from ''u'' to ''v''. Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length 1, i.e. by a single edge, the vertices are called adjacent. A [[Graph (mathematics)|graph]] is said to be connected if every pair of vertices in the graph is connected.
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| A [[connected component (graph theory)|connected component]] is a maximal connected subgraph of ''G''. Each vertex belongs to exactly one connected component, as does each edge.
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| A [[directed graph]] is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph.
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| It is connected if it contains a directed path from ''u'' to ''v'' or a directed path from ''v'' to ''u'' for every pair of vertices ''u'', ''v''. It is strongly connected or strong if it contains a directed path from ''u'' to ''v'' and a directed path from ''v'' to ''u'' for every pair of vertices ''u'', ''v''. The [[strongly connected component|strong components]] are the maximal strongly connected subgraphs.
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| A [[Cut (graph theory)|cut]], vertex cut, or separating set of a connected graph ''G'' is a set of vertices whose removal renders ''G'' disconnected. The connectivity or [[K-vertex-connected graph|vertex connectivity]] κ(''G'') (where G is not a [[complete graph]]) is the size of a minimal vertex cut. A graph is called ''k''-connected or ''k''-vertex-connected if its vertex connectivity is ''k'' or greater. This means a graph G is said to be k-connected if there does not exist a set of k-1 vertices whose removal disconnects the graph. A [[complete graph]] with ''n'' vertices, denoted <math>K_n</math>, has no vertex cuts at all, but by convention κ(''<math>K_n</math>'') = ''n''-1. A vertex cut for two vertices ''u'' and ''v'' is a set of vertices whose removal from the graph disconnects ''u'' and ''v''. The local connectivity κ(''u'', ''v'') is the size of a smallest vertex cut separating ''u'' and ''v''. Local connectivity is symmetric for undirected graphs; that is, κ(''u'',''v'')=κ(''v'',''u''). Moreover, except for complete graphs, κ(''G'') equals the minimum of κ(''u'',''v'') over all nonadjacent pairs of vertices ''u'', ''v''.
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| 2-connectivity is also called [[biconnected graph|biconnectivity]] and 3-connectivity is also called triconnectivity.
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| Analogous concepts can be defined for edges. In the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a [[bridge (graph theory)|bridge]]. More generally, the edge cut of ''G'' is a group of edges whose total removal renders the graph disconnected. The [[edge-connectivity]] λ(''G'') is the size of a smallest edge cut, and the local edge-connectivity λ(''u'',''v'') of two vertices ''u'', ''v'' is the size of a smallest edge cut disconnecting ''u'' from ''v''. Again, local edge-connectivity is symmetric. A graph is called ''k''-edge-connected if its edge connectivity is ''k'' or greater.
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| ==Menger's theorem==
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| {{main|Menger's theorem}}
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| One of the most important facts about connectivity in graphs is [[Menger's theorem]], which characterizes the connectivity and edge-connectivity of a graph in terms of the number of independent paths between vertices.
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| If ''u'' and ''v'' are vertices of a graph ''G'', then a collection of paths between ''u'' and ''v'' is called independent if no two of them share a vertex (other than ''u'' and ''v'' themselves). Similarly, the collection is edge-independent if no two paths in it share an edge. The number of mutually independent paths between ''u'' and ''v'' is written as κ′(''u'',''v''), and the number of mutually edge-independent paths between ''u'' and ''v'' is written as λ′(''u'',''v'').
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| Menger's theorem asserts that the local connectivity κ(''u'',''v'') equals κ′(''u'',''v'') and the local edge-connectivity λ(''u'',''v'') equals λ′(''u'',''v'') for every pair of vertices ''u'' and ''v''.<ref>{{cite book|title=Algorithmic Graph Theory|author=Gibbons, A.|year=1985|publisher=[[Cambridge University Press]]}}</ref><ref>{{cite book|title=Algorithmic Aspects of Graph Connectivity|author=Nagamochi, H., Ibaraki, T.|year=2008|publisher=Cambridge University Press}}</ref> This fact is actually a special case of the [[max-flow min-cut theorem]].
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| ==Computational aspects==
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| The problem of determining whether two vertices in a graph are connected can be solved efficiently using a [[search algorithm]], such as [[breadth-first search]]. More generally, it is easy to determine computationally whether a graph is connected (for example, by using a [[Disjoint-set data structure#Applications|disjoint-set data structure]]), or to count the number of connected components. A simple algorithm might be written in [[pseudo-code]] as follows:
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| #Begin at any arbitrary node of the graph, G
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| #Proceed from that node using either depth-first or breadth-first search, counting all nodes reached.
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| #Once the graph has been entirely traversed, if the number of nodes counted is equal to the number of nodes of G, the graph is connected; otherwise it is disconnected.
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| By Menger's theorem, for any two vertices ''u'' and ''v'' in a connected graph ''G'', the numbers κ(''u'',''v'') and λ(''u'',''v'') can be determined efficiently using the [[Max flow min cut|max-flow min-cut]] algorithm. The connectivity and edge-connectivity of ''G'' can then be computed as the minimum values of κ(''u'',''v'') and λ(''u'',''v''), respectively.
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| In computational complexity theory, [[SL (complexity)|SL]] is the class of problems [[log-space reducible]] to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to [[L (complexity)|L]] by [[Omer Reingold]] in 2004.<ref>{{Cite journal|first=Omer|last=Reingold|authorlink=Omer Reingold|title=Undirected connectivity in log-space|journal=[[Journal of the ACM]]|year=2008|volume=55|issue=4|pages=Article 17, 24 pages|doi=10.1145/1391289.1391291|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| Hence, undirected graph connectivity may be solved in <math>O(\log n)</math> space.
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| The problem of computing the probability that a [[Bernoulli distribution|Bernoulli]] random graph is connected is called Network reliability and the problem of computing whether two given vertices are connected the ST-reliability problem. Both of these are [[sharp-P|#P]]-hard{{Citation needed|date=April 2013}}. | |
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| ==Examples==
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| * The vertex- and edge-connectivities of a disconnected graph are both 0.
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| * 1-connectedness is synonymous with connectedness.
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| * The [[complete graph]] on ''n'' vertices has edge-connectivity equal to ''n'' − 1. Every other simple graph on ''n'' vertices has strictly smaller edge-connectivity.
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| * In a [[tree (graph theory)|tree]], the local edge-connectivity between every pair of vertices is 1.
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| == Bounds on connectivity ==
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| * The vertex-connectivity of a graph is less than or equal to its edge-connectivity. That is, κ(G) ≤ λ(G). Both are less than or equal to the [[Degree (graph theory)|minimum degree]] of the graph, since deleting all neighbors of a vertex of minimum degree will disconnect that vertex from the rest of the graph.<ref name="diestel">Diestel, R., [http://diestel-graph-theory.com/GrTh.html Graph Theory, Electronic Edition], 2005, p 12.</ref>
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| * For a [[vertex-transitive graph]] of [[Degree (graph theory)|degree]] ''d'', we have: 2(''d''+1)/3 ≤ κ(G) ≤ λ(G) = ''d''.<ref name="GandR">{{cite book|title=Algebraic Graph Theory|last1=Godsil|first1=C.|author1-link=Chris Godsil|last2=Royle|first2=G.|author2-link=Gordon Royle|year=2001|publisher=Springer Verlag}}</ref>
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| * For a [[vertex-transitive graph]] of [[Degree (graph theory)|degree]] ''d'' ≤ 4, or for any (undirected) minimal [[Cayley graph]] of [[Degree (graph theory)|degree]] ''d'', or for any [[symmetric graph]] of [[Degree (graph theory)|degree]] ''d'', both kinds of connectivity are equal: κ(G) = λ(G) = ''d''.<ref>{{cite book|title=Automorphism groups, isomorphism, reconstruction|series=Technical Report TR-94-10|author=Babai, L.|authorlink=László Babai|year=1996|publisher=University of Chicago|url=http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps}} Chapter 27 of ''The Handbook of Combinatorics''.</ref>
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| == Other properties ==
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| * Connectedness is preserved by [[graph homomorphism]]s.
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| * If ''G'' is connected then its [[line graph]] ''L''(''G'') is also connected.
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| * A graph ''G'' is 2-edge-connected [[if and only if]] it has an orientation that is strongly connected.
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| * [[Balinski's theorem]] states that the [[polytopal graph]] (1-[[Skeleton (topology)|skeleton]]) of a <math>k</math>-dimensional convex [[polytope]] is a <math>k</math>-vertex-connected graph.<ref>{{cite journal|author = Balinski, M. L.|title = On the graph structure of convex polyhedra in ''n''-space|journal = [[Pacific Journal of Mathematics]]|volume = 11|issue = 2|year = 1961| pages = 431–434|url = http://www.projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103037323}}</ref> As a partial converse, [[Ernst Steinitz|Steinitz]] showed that any 3-vertex-connected [[planar graph]] is a polytopal graph ([[Steinitz theorem]]).
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| * According to a theorem of [[Gabriel Andrew Dirac|G. A. Dirac]], if a graph is ''k''-connected for ''k'' ≥ 2, then for every set of ''k'' vertices in the graph there is a cycle that passes through all the vertices in the set.<ref>{{Cite journal
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| | last = Dirac | first = Gabriel Andrew | authorlink = Gabriel Andrew Dirac
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| | journal = [[Mathematische Nachrichten]]
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| | pages = 61–85
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| | title = In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen
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| | volume = 22
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| | year = 1960
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| | mr = 0121311
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| | doi = 10.1002/mana.19600220107
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref><ref>{{Cite journal
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| | last1 = Flandrin | first1 = Evelyne
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| | last2 = Li | first2 = Hao
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| | last3 = Marczyk | first3 = Antoni
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| | last4 = Woźniak | first4 = Mariusz
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| | doi = 10.1016/j.disc.2005.11.052
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| | issue = 7–8
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| | journal = Discrete Mathematics
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| | pages = 878–884
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| | title = A generalization of Dirac's theorem on cycles through ''k'' vertices in ''k''-connected graphs
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| | volume = 307
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| | year = 2007
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| | mr = 2297171
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref> The converse is true when ''k'' = 2.
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| == See also ==
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| * [[Algebraic connectivity]]
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| * [[Cheeger constant (graph theory)]]
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| * [[Expander graph]]
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| * [[Graph property]]
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| * [[Scale-free network]]
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| * [[Small-world networks]], [[Six degrees of separation]], [[Small world phenomenon]]
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| * [[Strength of a graph (graph theory)]]
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| ==References==
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| {{reflist}}
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| [[Category:Graph connectivity| ]]
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During the Neolithic Revolution, humans started consuming large amounts of dairy products, beans, cereals, alcohol and salt. This is an important point to be notices that the paleo diet menu was so restrictive and was without any taste as it had a less variety in it. Trail mix ideal snack for breakfast, lunch or dinner. There is no reason to measure portions since the the amount of protein in the diet fills you up and satisfies your appetite, unlike processed foods with their empty carbs and sugar which just make you hungrier and thirstier making it easy to eat too much. During this little bit of facts, we might end up being concentrating on Paleo diet plan, this is a common diet regime technique pertaining to reducing body fat, increasing muscle tissues, stopping center conditions, and also increasing cholestrerol levels.
Sources of potassium are beans, legumes, nuts, dairy products, fruits and vegetables. In order to do well in all ten of the fitness domains, your body must have real food to burn. There are certain instances when one particular desires to create some meals suing a swift recipe with fewer components then the paleo diet program menu recipes are the ideal option so we can't say that we are able to under no circumstances think about these recipes as they may be beneficial at occasions. Advocates of the Paleo diet plan motives that historical gentlemen are considerably more more healthy than contemporary adult males predominantly due to what they take in. Evolution has not caught up with our present day millisecond mentality.
Store in refrigerator in opaque air-tight container. Almonds can help to reduce cholesterol, lessen the likelihood of cardiac arrest and are also acknowledged with reducing the risk of several malignancies such as breast malignancies. However many others take into consideration Paleo Diet Eating Plan is pretty demanding, it has 1000's of followers. The human body needs a certain amount and quality of fat for certain processes. Listed here are a few ways to help bring you on the right path:.
Enjoy Paleo Chicken Stir Fry Recipe and Paleo Beef Spinach Salad Recipes. Not an expensive weight loss plan, no surgery and will not starve. It is the indigestible component in certain foods such as most fruits and vegetables which work to clean our digestive system so that we are able to allow only healthy substances into our bodies through our bloodstream. To lose that stubborn belly fat though, you've got to eliminate dairy and grain based products from your diet. He ate fruits and vegetables, which grew in abundance.
Sunflower seeds, pumpkin seeds and flax seeds all offer healthy proteins. The sad part is, most of us don't even know that we're eating that much. It's a lesson to those who think that going gluten-free automatically means weight loss. Day one - cabbage soup and all sorts of fruits except banana. Although there is no local 'Paleo Club' location for you to go to, many websites have been built to provide support and a community for people choosing to eat Paleo.
The Paleo diet reduces body fat, intensifies the immune system and increase your output of energy. Here are some items that you will surely love about this. A weakened jawbone can cause teeth to loosen, leaving you at a greater risk for gum disease. Would you comprehend the information as well as facts regarding the foodstuff you make. Some of the most devoted Paleolithic adherents also adopt a workout regimen that is thought to be based more closely on the workouts of our ancestors.