Induced subgraph in the context of Component (graph theory)

⭐ Core Definition: Induced subgraph

In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges, from the original graph, connecting pairs of vertices in that subset.

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👉 Induced subgraph in the context of Component (graph theory)

In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components.

The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not.

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⭐ Core Definition: Induced subgraph
👉 Induced subgraph in the context of Component (graph theory)
Induced subgraph in the context of Neighbourhood (graph theory)

Induced subgraph in the context of Neighbourhood (graph theory)

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 $v$ .

The neighbourhood is often denoted ⁠ $N_{G}(v)$ ⁠ 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 ⁠ $N_{G}[v]$ ⁠. When stated without any qualification, a neighbourhood is assumed to be open.