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| {{Graph families defined by their automorphisms}}
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| In [[graph theory]], a branch of mathematics, a '''skew-symmetric graph''' is a [[directed graph]] that is [[graph isomorphism|isomorphic]] to its own [[transpose graph]], the graph formed by reversing all of its edges, under an isomorphism that is an [[involution (mathematics)|involution]] without any [[Fixed point (mathematics)|fixed points]]. Skew-symmetric graphs are identical to the [[Bipartite double cover|double covering graphs]] of [[bidirected graph]]s.
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| Skew-symmetric graphs were first introduced under the name of ''antisymmetrical digraphs'' by {{harvtxt|Tutte|1967}}, later as the double covering graphs of polar graphs by {{harvtxt|Zelinka|1976b}}, and still later as the double covering graphs of bidirected graphs by {{harvtxt|Zaslavsky|1991}}. They arise in modeling the search for alternating paths and alternating cycles in algorithms for finding [[Matching (graph theory)|matchings]] in graphs, in testing whether a [[still life (cellular automaton)|still life]] pattern in [[Conway's Game of Life]] may be partitioned into simpler components, in [[graph drawing]], and in the [[implication graph]]s used to efficiently solve the [[2-satisfiability]] problem.
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| ==Definition==
| | Surely the second option would be more beneficial for any website. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>These websites can be easily customized and can appear in the top rankings of the major search engines. If you have any sort of questions relating to where and exactly how to make use of [http://xyz.ms/wordpress_dropbox_backup_715115 wordpress backup plugin], you could contact us at our own webpage. Infertility can cause a major setback to the couples due to the inability to conceive. Several claim that Wordpress just isn't an preferred tool to utilise when developing a professional site. Now, I want to anxiety that not every single query will be answered. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>ve labored so hard to publish and put up on their website. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Article Source: Stevens works in Internet and Network Marketing. <br><br>You can add keywords but it is best to leave this alone. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. Contact Infertility Clinic Providing One stop Fertility Solutions at:. Make sure you have the latest versions of all your plugins are updated. <br><br>Website security has become a major concern among individuals all over the world. An ease of use which pertains to both internet site back-end and front-end users alike. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. 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. |
| As defined, e.g., by {{harvtxt|Goldberg|Karzanov|1996}}, a skew-symmetric graph ''G'' is a directed graph, together with a function σ mapping vertices of ''G'' to other vertices of ''G'', satisfying the following properties:
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| # For every vertex ''v'', σ(''v'') ≠ ''v'',
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| # For every vertex ''v'', σ(σ(''v'')) = ''v'',
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| # For every edge (''u'',''v''), (σ(''v''),σ(''u'')) must also be an edge.
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| One may use the third property to extend σ to an orientation-reversing function on the edges of ''G''.
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| The [[transpose graph]] of ''G'' is the graph formed by reversing every edge of ''G'', and σ defines a [[graph isomorphism]] from ''G'' to its transpose. However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism. | |
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| A path or cycle in a skew-symmetric graph is said to be ''regular'' if, for each vertex ''v'' of the path or cycle, the corresponding vertex σ(''v'') is not part of the path or cycle.
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| ==Examples==
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| Every directed [[path graph]] with an even number of vertices is skew-symmetric, via a symmetry that swaps the two ends of the path. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation-reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs.
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| Similarly, a directed [[cycle graph]] is skew-symmetric if and only if it has an even number of vertices. In this case, the number of different mappings σ that realize the skew symmetry of the graph equals half the length of the cycle.
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| ==Polar/switch graphs, double covering graphs, and bidirected graphs==
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| A skew-symmetric graph may equivalently be defined as the double covering graph of a ''polar graph'' (introduced by {{harvtxt|Zelinka|1974}}, {{harvtxt|Zelinka|1976}}, called a ''switch graph'' by {{harvtxt|Cook|2003}}), which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. Each vertex of the polar graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph corresponds to two edges of the skew-symmetric graph. This equivalence is the one used by {{harvtxt|Goldberg|Karzanov|1996}} to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges. Zelinka (following F. Zitek) and Cook visualize the vertices of a polar graph as points where multiple tracks of a [[train track (mathematics)|train track]] come together: if a train enters a switch via a track that comes in from one direction, it must exit via a track in the other direction. The problem of finding non-self-intersecting smooth curves between given points in a train track comes up in testing whether certain kinds of [[graph drawing]]s are valid {{harv|Hui|Schaefer|Štefankovič|2004}} and may be modeled as the search for a regular path in a skew-symmetric graph.
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| A closely related concept is the [[bidirected graph]] of {{harvtxt|Edmonds|Johnson|1970}} ("polarized graph" in the terminology of {{harvtxt|Zelinka|1974}}, {{harvtxt|Zelinka|1976}}), a graph in which each of the two ends of each edge may be either a head or a tail, independently of the other end. A bidirected graph may be interpreted as a polar graph by letting the partition of edges at each vertex be determined by the partition of endpoints at that vertex into heads and tails; however, swapping the roles of heads and tails at a single vertex ("switching" the vertex, in the terminology of {{harvtxt|Zaslavsky|1991}}) produces a different bidirected graph but the same polar graph.
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| For the correspondence between bidirected graphs and skew-symmetric graphs (i.e., their double covering graphs) see {{harvtxt|Zaslavsky|1991}}, Section 5, or {{harvtxt|Babenko|2006}}. | |
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| To form the double covering graph (i.e., the corresponding skew-symmetric graph) from a polar graph ''G'', create for each vertex ''v'' of ''G'' two vertices ''v''<sub>0</sub> and ''v''<sub>1</sub>, and let σ(''v''<sub>''i''</sub>) = ''v''<sub>1 − ''i''</sub>. For each edge ''e'' = (''u'',''v'') of ''G'', create two directed edges in the covering graph, one oriented from ''u'' to ''v'' and one oriented from ''v'' to ''u''. If ''e'' is in the first subset of edges at ''v'', these two edges are from ''u''<sub>0</sub> into ''v''<sub>0</sub> and from ''v''<sub>1</sub> into ''u''<sub>1</sub>, while if ''e'' is in the second subset, the edges are from ''u''<sub>0</sub> into ''v''<sub>1</sub> and from ''v''<sub>0</sub> into ''u''<sub>1</sub>.
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| In the other direction, given a skew-symmetric graph ''G'', one may form a polar graph that has one vertex for every corresponding pair of vertices in ''G'' and one undirected edge for every corresponding pair of edges in ''G''. The undirected edges at each vertex of the polar graph may be partitioned into two subsets according to which vertex of the polar graph they go out of and come in to.
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| A regular path or cycle of a skew-symmetric graph corresponds to a path or cycle in the polar graph that uses at most one edge from each subset of edges at each of its vertices.
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| ==Matching==
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| In constructing [[Matching (graph theory)|matchings]] in undirected graphs, it is important to find ''alternating paths'', paths of vertices that start and end at unmatched vertices, in which the edges at odd positions in the path are not part of a given partial matching and in which the edges at even positions in the path are part of the matching. By removing the matched edges of such a path from a matching, and adding the unmatched edges, one can increase the size of the matching. Similarly, cycles that alternate between matched and unmatched edges are of importance in weighted matching problems.
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| As {{harvtxt|Goldberg|Karzanov|1996}} showed, an alternating path or cycle in an undirected graph may be modeled as a regular path or cycle in a skew-symmetric directed graph. To create a skew-symmetric graph from an undirected graph ''G'' with a specified matching ''M'', view ''G'' as a switch graph in which the edges at each vertex are partitioned into matched and unmatched edges; an alternating path in ''G'' is then a regular path in this switch graph and an alternating cycle in ''G'' is a regular cycle in the switch graph.
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| {{harvtxt|Goldberg|Karzanov|1996}} generalized alternating path algorithms to show that the existence of a regular path between any two vertices of a skew-symmetric graph may be tested in linear time. Given additionally a non-negative length function on the edges of the graph that assigns the same length to any edge ''e'' and to σ(''e''), the shortest regular path connecting a given pair of nodes in a skew-symmetric graph with ''m'' edges and ''n'' vertices may be tested in time O(''m'' log ''n''). If the length function is allowed to have negative lengths, the existence of a negative regular cycle may be tested in polynomial time.
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| Along with the path problems arising in matchings, skew-symmetric generalizations of the [[max-flow min-cut theorem]] have also been studied ({{harvnb|Goldberg|Karzanov|2004}}; {{harvnb|Tutte|1967}}).
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| ==Still life theory==
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| {{harvtxt|Cook|2003}} shows that a [[still life (cellular automaton)|still life pattern]] in [[Conway's Game of Life]] may be partitioned into two smaller still lifes if and only if an associated switch graph contains a regular cycle. As he shows, for switch graphs with at most three edges per vertex, this may be tested in polynomial time by repeatedly removing [[Bridge (graph theory)|bridges]] (edges the removal of which disconnects the graph) and vertices at which all edges belong to a single partition until no more such simplifications may be performed. If the result is an [[empty graph]], there is no regular cycle; otherwise, a regular cycle may be found in any remaining bridgeless component. The repeated search for bridges in this algorithm may be performed efficiently using a dynamic graph algorithm of {{harvtxt|Thorup|2000}}.
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| Similar bridge-removal techniques in the context of matching were previously considered by {{harvtxt|Gabow|Kaplan|Tarjan|1999}}.
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| ==Satisfiability==
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| [[File:Implication graph.svg|thumb|240px|An [[implication graph]]. Its skew symmetry can be realized by rotating the graph through a 180 degree angle and reversing all edges.]]
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| An instance of the [[2-satisfiability]] problem, that is, a Boolean expression in [[conjunctive normal form]] with two variables or negations of variables per clause, may be transformed into an [[implication graph]] by replacing each clause <math>\scriptstyle u\lor v</math> by the two implications
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| <math>\scriptstyle(\lnot u)\Rightarrow v</math> and <math>\scriptstyle(\lnot v)\Rightarrow u</math>. This graph has a vertex for each variable or negated variable, and a directed edge for each implication; it is, by construction, skew-symmetric, with a correspondence σ that maps each variable to its negation. | |
| As {{harvtxt|Aspvall|Plass|Tarjan|1979}} showed, a satisfying assignment to the 2-satisfiability instance is equivalent to a partition of this implication graph into two subsets of vertices, ''S'' and σ(''S''), such that no edge starts in ''S'' and ends in σ(''S''). If such a partition exists, a satisfying assignment may be formed by assigning a true value to every variable in ''S'' and a false value to every variable in σ(''S''). This may be done if and only if no [[strongly connected component]] of the graph contains both some vertex ''v'' and its complementary vertex σ(''v''). If two vertices belong to the same strongly connected component, the corresponding variables or negated variables are constrained to equal each other in any satisfying assignment of the 2-satisfiability instance. The total time for testing strong connectivity and finding a partition of the implication graph is linear in the size of the given 2-CNF expression.
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| ==Recognition==
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| It is [[NP-complete]] to determine whether a given directed graph is skew-symmetric, by a result of {{harvtxt|Lalonde|1981}} that it is NP-complete to find a color-reversing involution in a [[bipartite graph]]. Such an involution exists if and only if the directed graph given by [[Orientation (graph theory)|orienting]] each edge from one color class to the other is skew-symmetric, so testing skew-symmetry of this directed graph is hard. This complexity does not affect path-finding algorithms for skew-symmetric graphs, because these algorithms assume that the skew-symmetric structure is given as part of the input to the algorithm rather than requiring it to be inferred from the graph alone.
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| ==References==
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| *{{citation
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| | title = A linear-time algorithm for testing the truth of certain quantified boolean formulas
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| | contribution = Acyclic bidirected and skew-symmetric graphs: algorithms and structure
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| | contribution = Matching: a well-solved class of linear programs
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| | title = Combinatorial Structures and their Applications: Proceedings of the Calgary Symposium, June 1969
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| | publisher = Gordon and Breach | location = New York | year = 1970
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| }}. Reprinted in ''Combinatorial Optimization — Eureka, You Shrink!'', Springer-Verlag, Lecture Notes in Computer Science 2570, 2003, pp. 27–30, {{doi|10.1007/3-540-36478-1_3}}.
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| *{{citation
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| | doi = 10.1145/301250.301273
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| *{{citation
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| | last2 = Karzanov | first2 = Alexander V.
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| | title = Path problems in skew-symmetric graphs
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| | journal = Combinatorica
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| | title = Maximum skew-symmetric flows and matchings
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| | journal = Mathematical programming
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| *{{citation
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| | last2 = Schaefer | first2 = Marcus
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| | last3 = Štefankovič | first3 = Daniel
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| | contribution = Train tracks and confluent drawings
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| | publisher = Springer-Verlag
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| *{{citation
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| | title = Le problème d'étoiles pour graphes est NP-complet
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| *{{citation|first=Mikkel|last=Thorup|authorlink=Mikkel Thorup|year=2000|pages=343–350|doi=10.1145/335305.335345|contribution=Near-optimal {{Sic|hide=y|fully|-}}dynamic graph connectivity|title=[[Symposium on Theory of Computing|Proc. 32nd ACM Symposium on Theory of Computing]]|isbn=1-58113-184-4}}.
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| *{{citation
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| | journal = Canadian Journal of Mathematics
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| | volume = 19 | pages = 1101–1117 | year = 1967
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| | doi = 10.4153/CJM-1967-101-8}}.
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| *{{citation
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| *{{citation
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| | volume = 26 | pages = 352–360 | year = 1976b }}.
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| [[Category:Graph families]]
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| [[Category:Directed graphs]]
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| [[Category:Matching]]
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