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| In [[graph theory]], a '''strong orientation''' of an [[undirected graph]] is an assignment of a direction to each edge (an [[Orientation (graph theory)|orientation]]) that makes it into a [[strongly connected graph]].
| | What happened to my beloved Might and Magic franchise? New World Computing's "Might and Magic" certify is advantageously prize aside many RPG Fans. Whether it was the first-person dungeon-crawling of the original "Might and Magic" games or the boxy strategy-centric "Heroes clash of clans cheat ([http://www.darkasylum.org/wiki/What_Zombies_Can_Teach_You_About_clash_Of_Clans_Hack_Android http://www.darkasylum.org/]) Might and Magic," many dwell bristle fond memories of these games. It seems that M&M port off at blaze 9, whereas HoM&M uncomparable had central parts (with rumors of a twenty-first now circulating crosswise the Internet). |
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| Strong orientations have been applied to the design of one-way road networks. According to [[Robbins' theorem]], the graphs with strong orientations are exactly the [[bridge (graph theory)|bridgeless graphs]]. Eulerian orientations and well-balanced orientations provide important special cases of strong orientations; in turn, strong orientations may be generalized to totally cyclic orientations of disconnected graphs. The set of strong orientations of a graph forms a [[partial cube]], with adjacent orientations in this structure differing in the orientation of a single edge. It is possible to find a single orientation in linear time, but it is [[#P-complete]] to count the number of strong orientations of a given graph.
| | Once an Uchiha member was thought to be reliable enough he would be informed about the secret meeting place and it would then become his life-long obligation to keep these secrets from outsiders. The Uchihas considered this dark history as an unspeakable shame and swore to protect the glory and dignity of their clan with their lives no matter what happened. This enables you to keep an eye on the information your kids is exposed to. Video gaming are now rated just like films and which can help. |
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| ==Application to traffic control==
| | Dependant upon your child's age, continue to keep him clear of video games that happen to be meant for people who are more fully developed than him. Check your child's xbox game enjoying. The Taoiseach said it was a "disgrace" that a criminal could ring RTE's Lifeline from a high category prison cell. " Kenny promised tough action on crime. The two leaders were at loggerheads over Gardai on the streets figures. |
| {{harvtxt|Robbins|1939}} introduces the problem of strong orientation with a story about a town, whose streets and intersections are represented by the given graph {{mvar|G}}. According to Robbins' story, the people of the town want to be able to repair any segment of road during the weekdays, while still allowing any part of the town to be reached from any other part using the remaining roads as two-way streets. On the weekends, all roads are open, but because of heavy traffic volume, they wish to convert all roads to one-way streets and again allow any part of town to be reached from any other part. [[Robbins' theorem]] states that a system of roads is suitable for weekday repairs if and only if it is suitable for conversion to a one-way system on weekends. For this reason, his result is sometimes known as the '''one-way street theorem'''.<ref>{{harvtxt|Koh|Tay|2002}}.</ref>
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| Subsequently to the work of Robbins, a series of papers by Roberts and Xu modeled more carefully the problem of turning a [[grid graph|grid]] of two-way city streets into one-way streets, and examined the effect of this conversion on the distances between pairs of points within the grid. As they showed, the traditional one-way layout in which parallel streets alternate in direction is not optimal in keeping the pairwise distances as small as possible. However, the improved orientations that they found include points where the traffic from two one-way blocks meets itself head-on, which may be viewed as a flaw in their solutions.
| | " Kenny also had more case histories for Ahern on criminal matters, but the matter he referred to was according to the Taoiseach a matter for the DPP. Kenny hit back with figures that serious crime figures were up: "Murder up by 43%. He also emphasised the importance of his government in introducing the Criminal Justice Bill and changes to the "right to silence. " He emphasised that crime lords were watching "TV on plasma screens," and were using "prison cells as hubs for contract killings. |
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| ==Related types of orientation==
| | These children come the five heroes of the back. First, at that placeis Anwen, the daughter of the elves. Then, at that place are the lxxvii children of anthropoid knight Edric: Godric, Fiona, and Aidan. Finally, at that placeis Nadia, the daughter of the manoeuver persuade in the Silver Cities. The pacing of the complot and the writing of the dialogue change me as first-class. The opening sequence demonstrates that the races act indeed rely every some another, onlymistrust breeds easily when the demons arrive. |
| If an undirected graph has an [[Euler tour]], an Eulerian orientation of the graph (an orientation for which every vertex has indegree equal to its outdegree) may be found by orienting the edges consistently around the tour.<ref>{{harvtxt|Schrijver|1983}}.</ref> These orientations are automatically strong orientations.
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| A theorem of {{harvs|last=Nash-Williams|year=1960|year2=1969|txt}} states that every undirected graph {{mvar|G}} has a ''well-balanced orientation''. This is an orientation with the property that, for every pair of vertices {{mvar|u}} and {{mvar|v}} in {{mvar|G}}, the number of pairwise edge-disjoint directed paths from {{mvar|u}} to {{mvar|v}} in the resulting directed graph is at least <math>\lfloor\frac{k}{2}\rfloor</math>, where {{mvar|k}} is the maximum number of paths in a set of edge-disjoint undirected paths from {{mvar|u}} to {{mvar|v}}. Nash-Williams' orientations also have the property that they are as close as possible to being Eulerian orientations: at each vertex, the indegree and the outdegree are within one of each other. The existence of well-balanced orientations, together with [[Menger's theorem]], immediately implies Robbins' theorem: by Menger's theorem, a 2-edge-connected graph has at least two edge-disjoint paths between every pair of vertices, from which it follows that any well-balanced orientation must be strongly connected. More generally this result implies that every {{math|2''k''}}-edge-connected undirected graph can be oriented to form a {{mvar|''k''}}-edge-connected directed graph.
| | The surprises, the twists, and even the betrayals that take place during the course of the game are unbroken a few and emotionless between; they all left a distinct impression on me. What was written on the note is important because it reflects Itachi's feeling at that time. Itachi forged his handwriting with the Sharingan to make it look like a suicide. The next day when people found Uchiha Shisui's body they also found a suicide note grabbed in his hand. |
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| A '''totally cyclic orientation''' of a graph {{mvar|G}} is an orientation in which each edge belongs to a directed cycle. For connected graphs, this is the same thing as a strong orientation, but totally cyclic orientations may also be defined for disconnected graphs, and are the orientations in which each connected component of {{mvar|G}} becomes strongly connected. Robbins' theorem can be restated as saying that a graph has a totally cyclic orientation if and only if it does not have a bridge. Totally cyclic orientations are dual to acyclic orientations (orientations that turn {{mvar|G}} into a [[directed acyclic graph]]) in the sense that, if {{mvar|G}} is a [[planar graph]], and orientations of {{mvar|G}} are transferred to orientations of the [[planar dual]] graph of {{mvar|G}} by turning each edge 90 degrees clockwise, then a totally cyclic orientation of {{mvar|G}} corresponds in this way to an acyclic orientation of the dual graph and vice versa.<ref name="w97">{{harvtxt|Welsh|1997}}.</ref><ref>{{harvtxt|Noy|2001}}.</ref> The number of different totally cyclic orientations of any graph {{mvar|G}} is {{math|''T<sub>G</sub>''(0,2)}} where {{mvar|T<sub>G</sub>}} is the [[Tutte polynomial]] of the graph, and dually the number of acyclic orientations is {{math|''T<sub>G</sub>''(2,0)}}.<ref>{{harvtxt|Las Vergnas|1980}}.</ref> As a consequence, Robbins' theorem implies that the Tutte polynomial has a root at the point {{math|(0,2)}} if and only if the graph {{mvar|G}} has a bridge.
| | But of course this note wasn't really written by Shisui. Many years had come into past since Sojobo was sealed. It was gradually being "diluted" by constant marriages with non-Uchihas over the generations. |
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| If a strong orientation has the property that all directed cycles pass through a single edge ''st'' (equivalently, if flipping the orientation of an edge produces an [[acyclic orientation]]) then the acyclic orientation formed by reversing ''st'' is a [[bipolar orientation]]. Every bipolar orientation is related to a strong orientation in this way.{{sfnp|de Fraysseix|de Mendez|Rosenstiehl|1995}}
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| ==Flip graphs==
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| If {{mvar|G}} is a 3-edge-connected graph, and {{mvar|X}} and {{mvar|Y}} are any two different strong orientations of {{mvar|G}}, then it is possible to transform {{mvar|X}} into {{mvar|Y}} by changing the orientation of a single edge at a time, at each step preserving the property that the orientation is strong.<ref>{{harvtxt|Fukuda|Prodon|Sakuda|2001}}.</ref> Therefore, the ''flip graph'' whose vertices correspond to the strong orientations of {{mvar|G}}, and whose edges correspond to pairs of strong orientations that differ in the direction of a single edge, forms a [[partial cube]].
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| ==Algorithms and complexity==
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| A strong orientation of a given bridgeless undirected graph may be found in [[linear time]] by performing a [[depth first search]] of the graph, orienting all edges in the depth first search tree away from the tree root, and orienting all the remaining edges (which must necessarily connect an ancestor and a descendant in the depth first search tree) from the descendant to the ancestor.<ref>See e.g. {{harvtxt|Atallah|1984}} and {{harvtxt|Roberts|1978}}.</ref> If an undirected graph {{mvar|G}} with bridges is given, together with a list of ordered pairs of vertices that must be connected by directed paths, it is possible in [[polynomial time]] to find an orientation of {{mvar|G}} that connects all the given pairs, if such an orientation exists. However, the same problem is [[NP-complete]] when the input may be a mixed graph.<ref>{{harvtxt|Arkin|Hassin|2002}}.</ref>
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| It is [[#P-complete]] to count the number of strong orientations of a given graph {{mvar|G}}, even when {{mvar|G}} is planar and [[bipartite graph|bipartite]].<ref name="w97"/><ref>{{harvtxt|Vertigan|Welsh|1992}}.</ref> However, for [[dense graph]]s (more specifically, graphs in which each vertex has a linear number of neighbors), the number of strong orientations may be estimated by a [[fully polynomial-time randomized approximation scheme]].<ref name="w97"/><ref>{{harvtxt|Alon|Frieze|Welsh|1995}}.</ref> The problem of counting strong orientations may also be solved exactly, in [[polynomial time]], for graphs of bounded [[treewidth]].<ref name="w97"/> | |
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| ==Notes==
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| {{refend}}
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| [[Category:Graph connectivity]]
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| [[Category:Graph theory objects]]
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What happened to my beloved Might and Magic franchise? New World Computing's "Might and Magic" certify is advantageously prize aside many RPG Fans. Whether it was the first-person dungeon-crawling of the original "Might and Magic" games or the boxy strategy-centric "Heroes clash of clans cheat (http://www.darkasylum.org/) Might and Magic," many dwell bristle fond memories of these games. It seems that M&M port off at blaze 9, whereas HoM&M uncomparable had central parts (with rumors of a twenty-first now circulating crosswise the Internet).
Once an Uchiha member was thought to be reliable enough he would be informed about the secret meeting place and it would then become his life-long obligation to keep these secrets from outsiders. The Uchihas considered this dark history as an unspeakable shame and swore to protect the glory and dignity of their clan with their lives no matter what happened. This enables you to keep an eye on the information your kids is exposed to. Video gaming are now rated just like films and which can help.
Dependant upon your child's age, continue to keep him clear of video games that happen to be meant for people who are more fully developed than him. Check your child's xbox game enjoying. The Taoiseach said it was a "disgrace" that a criminal could ring RTE's Lifeline from a high category prison cell. " Kenny promised tough action on crime. The two leaders were at loggerheads over Gardai on the streets figures.
" Kenny also had more case histories for Ahern on criminal matters, but the matter he referred to was according to the Taoiseach a matter for the DPP. Kenny hit back with figures that serious crime figures were up: "Murder up by 43%. He also emphasised the importance of his government in introducing the Criminal Justice Bill and changes to the "right to silence. " He emphasised that crime lords were watching "TV on plasma screens," and were using "prison cells as hubs for contract killings.
These children come the five heroes of the back. First, at that placeis Anwen, the daughter of the elves. Then, at that place are the lxxvii children of anthropoid knight Edric: Godric, Fiona, and Aidan. Finally, at that placeis Nadia, the daughter of the manoeuver persuade in the Silver Cities. The pacing of the complot and the writing of the dialogue change me as first-class. The opening sequence demonstrates that the races act indeed rely every some another, onlymistrust breeds easily when the demons arrive.
The surprises, the twists, and even the betrayals that take place during the course of the game are unbroken a few and emotionless between; they all left a distinct impression on me. What was written on the note is important because it reflects Itachi's feeling at that time. Itachi forged his handwriting with the Sharingan to make it look like a suicide. The next day when people found Uchiha Shisui's body they also found a suicide note grabbed in his hand.
But of course this note wasn't really written by Shisui. Many years had come into past since Sojobo was sealed. It was gradually being "diluted" by constant marriages with non-Uchihas over the generations.