# Neighbourhood (graph theory)

Last updated A graph consisting of 6 vertices and 7 edges
For other meanings of neighbourhoods in mathematics, see Neighbourhood (mathematics). For non-mathematical neighbourhoods, see Neighbourhood (disambiguation).

In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v. For example, in the image to the right, the neighbourhood of vertex 5 consists of vertices 1, 2 and 4 and the edge connecting vertices 1 and 2.

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The neighbourhood is often denoted NG(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 NG[v]. When stated without any qualification, a neighbourhood is assumed to be open.

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 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.

An isolated vertex has no adjacent vertices. The degree of a vertex is equal to the number of adjacent vertices. A special case is a loop that connects a vertex to itself; if such an edge exists, the vertex belongs to its own neighbourhood.

## Local properties in graphs In the octahedron graph, the neighbourhood of any vertex is a 4-cycle.

If all vertices in G have neighbourhoods that are 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 (Hell 1978, Sedláček 1983). For instance, in the octahedron graph, shown in the figure, each vertex has a neighbourhood isomorphic to a cycle of four vertices, so the octahedron is locally C4.

For example:

• Any complete graph Kn is locally Kn-1. The only graphs that are locally complete are disjoint unions of complete graphs.
• A Turán graph T(rs,r) is locally T((r-1)s,r-1). More generally any Turán graph is locally Turán.
• Every planar graph is locally outerplanar. However, not every locally outerplanar graph is planar.
• A graph is triangle-free if and only if it is locally independent.
• Every k-chromatic graph is locally (k-1)-chromatic. Every locally k-chromatic graph has chromatic number $O({\sqrt {kn}})$ ( Wigderson 1983 ).
• 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.
• A graph is locally cyclic if every neighbourhood is a cycle. For instance, the octahedron is the unique connected locally C4 graph, the icosahedron is the unique connected locally C5 graph, and the Paley graph of order 13 is locally C6. Locally cyclic graphs other than K4 are exactly the underlying graphs of Whitney triangulations, embeddings of graphs on surfaces in such a way that the faces of the embedding are the cliques of the graph (Hartsfeld & Ringel 1991; Larrión, Neumann-Lara & Pizaña 2002; Malnič & Mohar 1992). Locally cyclic graphs can have as many as $n^{2-o(1)}$ edges ( Seress & Szabó 1995 ).
• Claw-free graphs are the graphs that are locally co-triangle-free; that is, for all vertices, the complement graph of the neighbourhood 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 C5 and C5 has independence number two.
• The locally linear graphs are the graphs in which every neighbourhood is an induced matching ( Fronček 1989 ).

## Neighbourhood of a set

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.

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 graphs.

## Related Research Articles

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to Kelly and Ulam.

This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges. In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. The Turán graphT(n,r) is a complete multipartite graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge if and only if they belong to different subsets. The graph will have subsets of size , and subsets of size . That is, it is a complete r-partite graph In the mathematical area of graph theory, a clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs or triangulated graphs. In mathematics, topology generalizes the notion of triangulation in a natural way as follows: In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. In graph theory, a rook's graph is a graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and each edge represents a legal move from one square to another. The same graphs can be defined mathematically as the Cartesian products of two complete graphs or as the line graphs of complete bipartite graphs. In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. Clique complexes, flag complexes, and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques of an undirected graph. In the mathematical field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with 80 edges. The 80-edge variant is the order-5 halved cube graph; it was called the Clebsch graph name by Seidel (1968) because of its relation to the configuration of 16 lines on the quartic surface discovered in 1868 by the German mathematician Alfred Clebsch. The 40-edge variant is the order-5 folded cube graph; it is also known as the Greenwood–Gleason graph after the work of Robert E. Greenwood and Andrew M. Gleason (1955), who used it to evaluate the Ramsey number R(3,3,3) = 17. In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.

In graph theory, a mathematical discipline, coloring refers to an assignment of colours or labels to vertices, edges and faces of a graph. Defective coloring is a variant of proper vertex coloring. In a proper vertex coloring, the vertices are coloured such that no adjacent vertices have the same colour. In defective coloring, on the other hand, vertices are allowed to have neighbours of the same colour to a certain extent. In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph. Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length.

In graph theory, a universal vertex is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. In combinatorial mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs to a unique triangle. Equivalently it asks for the maximum number of edges in a balanced bipartite graph whose edges can be partitioned into a linear number of induced matchings, or the maximum number of triples one can choose from points so that every six points contain at most two triples. The problem is named after Imre Z. Ruzsa and Endre Szemerédi, who first proved that its answer is smaller than by a slowly-growing factor.

• Fronček, Dalibor (1989), "Locally linear graphs", Mathematica Slovaca, 39 (1): 3–6, hdl:10338.dmlcz/136481, MR   1016323
• Hartsfeld, Nora; Ringel, Gerhard (1991), "Clean triangulations", Combinatorica , 11 (2): 145–155, doi:10.1007/BF01206358 .
• Hell, Pavol (1978), "Graphs with given neighborhoods I", Problèmes combinatoires et théorie des graphes, Colloques internationaux C.N.R.S., 260, pp. 219–223.
• Larrión, F.; Neumann-Lara, V.; Pizaña, M. A. (2002), "Whitney triangulations, local girth and iterated clique graphs", Discrete Mathematics , 258 (1–3): 123–135, doi:10.1016/S0012-365X(02)00266-2 .
• Malnič, Aleksander; Mohar, Bojan (1992), "Generating locally cyclic triangulations of surfaces", Journal of Combinatorial Theory, Series B , 56 (2): 147–164, doi:.
• Sedláček, J. (1983), "On local properties of finite graphs", Graph Theory, Lagów, Lecture Notes in Mathematics, 1018, Springer-Verlag, pp. 242–247, doi:10.1007/BFb0071634, ISBN   978-3-540-12687-4 .
• Seress, Ákos; Szabó, Tibor (1995), "Dense graphs with cycle neighborhoods", Journal of Combinatorial Theory, Series B , 63 (2): 281–293, doi:10.1006/jctb.1995.1020, archived from the original on 2005-08-30.
• Wigderson, Avi (1983), "Improving the performance guarantee for approximate graph coloring", Journal of the ACM , 30 (4): 729–735, doi:10.1145/2157.2158 .