Neighbourhood (graph theory)

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A graph consisting of 6 vertices and 7 edges
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 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.

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.

[edit] Local properties in graphs

In the octahedron graph, the neighbourhood of any vertex is a 4-cycle.
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, Sedlacek 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 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 locally C4 graph, the icosahedron is the unique 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 and Ringel 1981; Larrión et al 2002; Malnič and Mohar 1992). Locally cyclic graphs can have as many as n2 − o(1) edges (Seress and Szabó 1995).

[edit] 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.

[edit] References

  • Hell, Pavol (1978). "Graphs with given neighborhoods I". Colloque internationaux C.N.R.S., No. 260, Problems Combinatories et theorie des graphes: 219–223. 
  • Malnič, Aleksander; Mohar, Bojan (1992). "Generating locally cyclic triangulations of surfaces". Journal of Combinatorial Theory, Series B 56 (2): 147–164. doi:10.1016/0095-8956(92)90015-P. 
  • Sedlacek, J. (1983). "On local properties of finite graphs". Graph Theory, Lagów: 242–247, Lecture Notes in Mathematics, no. 1018, Springer-Verlag. 
  • Wigderson, Avi (1983). "Improving the performance guarantee for approximate graph coloring". Journal of the ACM 30 (4): 729–735. doi:10.1145/2157.2158.