Component (graph theory) in the context of Matroid

Component (graph theory) in the context of Matroid

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⭐ Core Definition: 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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Component (graph theory) in the context of Distance (graph theory)

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.

In the case of a directed graph the distance d(u,v) between two vertices u and v is defined as the length of a shortest directed path from u to v consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, d(u,v) does not necessarily coincide with d(v,u)—so it is just a quasi-metric, and it might be the case that one is defined while the other is not.

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Graph theory → Undirected graph → Connected graph → Glossary of graph theory → Graph partition → Disjoint set → Induced subgraph → Graph invariant → Matroid → Topological space → Matrix (mathematics) → Random graph → Giant component → Percolation threshold →