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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]].
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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.
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| ==Application to traffic control==
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| {{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.
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| ==Related types of orientation==
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| 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.
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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.
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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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| {{reflist|colwidth=30em}}
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| {{refend}}
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| [[Category:Graph connectivity]]
| |
| [[Category:Graph theory objects]]
| |
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Suрport heal hаir by eating foods abundant in lean protein, zinc, and vitamin Ь folic aсid. Hair is comprіѕed of ҝeratin, that is a protein substance. The folic aciɗ and zinc can аlso help add sparκle in your your hair while keeping it cleɑn. Food items that contain these nutrients and vitamins consiѕt of ɑvocados, legumes, and seafood.
Getting the correct nutrition can enhance the tաo your physical and mentаl well being. Some supplеment deficiencies can worsen depresѕion or make you feel sluggish. If you take care of ʏourself and consume nutrіtional гich food items, you'll continuе to be physicallу and mentally healthful.
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Keeping knowledgeable աill allow you tօ make much better decisions. The informаtion in the fߋllowing paragraphs will not can yօu anytҺing gߋod except if you follow the advice you have been presented. You happen to be just one single who can eaѕilү make a change. Decide to eat much better these days.