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| :''For other meanings of neighbourhoods in mathematics, see [[Neighbourhood (mathematics)]]. For non-mathematical neighbourhoods, see [[Neighbourhood (disambiguation)]].''
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| In [[graph theory]], an '''adjacent vertex''' of a [[vertex (graph theory)|vertex]] ''v'' in a [[Graph (mathematics)|graph]] is a vertex that is connected to ''v'' by an [[edge (graph theory)|edge]]. The '''neighbourhood''' of a vertex ''v'' in a graph ''G'' is the [[induced subgraph#Subgraphs|induced subgraph]] of ''G'' consisting of all vertices adjacent to ''v'' and all edges connecting two such vertices. For example, the image shows a graph of 6 vertices and 7 edges. Vertex 5 is adjacent to vertices 1, 2, and 4 but it is not adjacent to 3 and 6. The neighbourhood of vertex 5 is the graph with three vertices, 1, 2, and 4, and one edge connecting vertices 1 and 2.
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| The neighbourhood is often denoted ''N''<sub>''G''</sub>(''v'') or (when the graph is unambiguous) ''N''(''v''). The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include ''v'' itself, and is more specifically the '''open neighbourhood''' of ''v''; it is also possible to define a neighbourhood in which ''v'' itself is included, called the '''closed neighbourhood''' and denoted by ''N''<sub>''G''</sub>[''v'']. When stated without any qualification, a neighbourhood is assumed to be open.
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| Neighbourhoods may be used to represent graphs in computer algorithms, via the [[adjacency list]] and [[adjacency matrix]] representations. Neighbourhoods are also used in the [[clustering coefficient]] of a graph, which is a measure of the average [[Dense graph|density]] of its neighbourhoods. In addition, many important classes of graphs may be defined by properties of their neighbourhoods, or by symmetries that relate neighbourhoods to each other.
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| An [[isolated vertex]] has no adjacent vertices. The [[degree (graph theory)|degree]] of a vertex is equal to the number of adjacent vertices. A special case is a [[loop (graph theory)|loop]] that connects a vertex to itself; if such an edge exists, the vertex belongs to its own neighbourhood.
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| ==Local properties in graphs==
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| [[Image:Octahedron graph.png|thumb|In the [[Octahedron|octahedron graph]], the neighbourhood of any vertex is a 4-[[Cycle graph|cycle]].]]
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| If all vertices in ''G'' have neighbourhoods that are [[Graph isomorphism|isomorphic]] to the same graph ''H'', ''G'' is said to be ''locally H'', and if all vertices in ''G'' have neighbourhoods that belong to some graph family ''F'', ''G'' is said to be ''locally F'' ({{harvnb|Hell|1978}}, {{harvnb|Sedlacek|1983}}). For instance, in the [[Octahedron|octahedron graph]] shown in the figure, each vertex has a neighbourhood isomorphic to a [[Cycle graph|cycle]] of four vertices, so the octahedron is locally ''C''<sub>4</sub>.
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| For example:
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| * Any [[complete graph]] ''K''<sub>''n''</sub> is locally ''K''<sub>''n-1''</sub>. The only graphs that are locally complete are disjoint unions of complete graphs.
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| * A [[Turán graph]] ''T''(''rs'',''r'') is locally ''T''((''r''-1)''s'',''r''-1). More generally any Turán graph is locally Turán.
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| * Every [[planar graph]] is locally [[Outerplanar graph|outerplanar]]. However, not every locally outerplanar graph is planar.
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| * A graph is [[triangle-free graph|triangle-free]] if and only if it is locally [[Independent set (graph theory)|independent]].
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| * Every ''k''-[[Chromatic number|chromatic]] graph is locally (''k''-1)-chromatic. Every locally ''k''-chromatic graph has chromatic number <math>O(\sqrt{kn})</math> {{harv|Wigderson|1983}}.
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| * If a graph family ''F'' is closed under the operation of taking induced subgraphs, then every graph in ''F'' is also locally ''F''. For instance, every [[chordal graph]] is locally chordal; every [[perfect graph]] is locally perfect; every [[comparability graph]] is locally comparable.
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| * A graph is locally cyclic if every neighbourhood is a [[Cycle graph|cycle]]. For instance, the [[octahedron]] is the unique locally ''C''<sub>4</sub> graph, the [[icosahedron]] is the unique locally ''C''<sub>5</sub> graph, and the [[Paley graph]] of order 13 is locally ''C''<sub>6</sub>. Locally cyclic graphs other than ''K''<sub>4</sub> are exactly the underlying graphs of [[Triangulation (topology)|Whitney triangulations]], embeddings of graphs on surfaces in such a way that the faces of the embedding are the cliques of the graph ({{harvnb|Hartsfeld|Ringel|1981}}; {{harvnb|Larrión|Neumann-Lara|Pizaña|2002}}; {{harvnb|Malnič|Mohar|1992}}). Locally cyclic graphs can have as many as <math>n^{2-o(1)}</math> edges {{harv|Seress|Szabó|1995}}.
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| * [[Claw-free graph]]s are the graphs that are locally co-[[triangle-free graph|triangle-free]]; that is, for all vertices, the [[complement graph]] of the neighborhood of the vertex does not contain a triangle. A graph that is locally ''H'' is claw-free if and only if the [[independence number]] of ''H'' is at most two; for instance, the graph of the regular icosahedron is claw-free because it is locally ''C''<sub>5</sub> and ''C''<sub>5</sub> has independence number two.
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| ==Neighbourhood of a Set==
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| For a set ''A'' of vertices, the neighbourhood of ''A'' is the union of the neighbourhoods of the vertices, and so it is the set of all vertices adjacent to at least one member of ''A''.
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| A set ''A'' of vertices in a graph is said to be a module if every vertex in ''A'' has the same set of neighbours outside of ''A''. Any graph has a uniquely recursive decomposition into modules, its [[modular decomposition]], which can be constructed from the graph in [[linear time]]; modular decomposition algorithms have applications in other graph algorithms including the recognition of [[comparability graph]]s.
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| ==See also==
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| * [[Moore neighborhood]]
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| * [[Von Neumann neighborhood]]
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| * [[Vertex figure]], a related concept in [[polyhedral geometry]]
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| ==References==
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| *{{citation
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| | last1 = Hartsfeld | first1 = Nora
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| | last2 = Ringel | first2 = Gerhard | author2-link = Gerhard Ringel
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| | doi = 10.1007/BF01206358
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| | journal = [[Combinatorica]]
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| | pages = 145–155
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| | title = Clean triangulations
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| | volume = 11
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| | year = 1991
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| | issue = 2}}.
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| *{{citation
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| | last = Hell | first = Pavol
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| | contribution = Graphs with given neighborhoods I
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| | pages = 219–223
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| | series = Colloque internationaux C.N.R.S.
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| | title = Problems Combinatories et theorie des graphes
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| | volume = 260
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| | year = 1978}}.
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| *{{citation
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| | last1 = Larrión | first1 = F.
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| | last2 = Neumann-Lara | first2 = V.
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| | last3 = Pizaña | first3 = M. A.
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| | doi = 10.1016/S0012-365X(02)00266-2
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | pages = 123–135
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| | title = Whitney triangulations, local girth and iterated clique graphs
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| | url = http://xamanek.izt.uam.mx/map/papers/cuello10_DM.ps
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| | volume = 258
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| | year = 2002}}.
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| *{{citation
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| | last1 = Malnič | first1 = Aleksander
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| | last2 = Mohar | first2 = Bojan | author2-link = Bojan Mohar
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| | doi = 10.1016/0095-8956(92)90015-P
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| | issue = 2
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| | journal = [[Journal of Combinatorial Theory|Journal of Combinatorial Theory, Series B]]
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| | pages = 147–164
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| | title = Generating locally cyclic triangulations of surfaces
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| | volume = 56
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| | year = 1992}}.
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| *{{citation
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| | last = Sedlacek | first = J.
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| | contribution = On local properties of finite graphs
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| | doi = 10.1007/BFb0071634
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| | pages = 242–247
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Mathematics
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| | title = Graph Theory, Lagów
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| | volume = 1018
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| | year = 1983
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| | chapter = On local properties of finite graphs
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| | isbn = 978-3-540-12687-4}}.
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| *{{citation
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| | last1 = Seress | first1 = Ákos
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| | last2 = Szabó | first2 = Tibor
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| | doi = 10.1006/jctb.1995.1020
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| | journal = [[Journal of Combinatorial Theory|Journal of Combinatorial Theory, Series B]]
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| | pages = 281–293
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| | title = Dense graphs with cycle neighborhoods
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| | url = http://www.inf.ethz.ch/personal/szabo/PS/kornyezetek.ps
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| | volume = 63
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| | year = 1995
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| | issue = 2}}.
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| *{{citation
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| | last = Wigderson | first = Avi | author-link = Avi Wigderson
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| | doi = 10.1145/2157.2158
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| | issue = 4
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| | journal = [[Journal of the ACM]]
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| | pages = 729–735
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| | title = Improving the performance guarantee for approximate graph coloring
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| | volume = 30
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| | year = 1983}}.
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| [[Category:Graph theory objects]]
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