Connected component

A connected component is a maximal set of vertices in a graph such that every pair is joined by a path. No vertex outside the set can be added without losing this property.

Graph theory: The mathematical study of graphs, structures made of vertices connected by edges. Connected components are a basic structural feature studied throughout graph theory.
Depth-first search: Depth-first search explores a graph by following one branch as far as possible before backtracking. A traversal from an unvisited vertex marks exactly the vertices in its component.
Social network analysis: Social network analysis studies relationships among people, groups, or organizations using network models. Components identify groups with no network path to the rest of the social graph.
Strongly connected component: A strongly connected component is a maximal set of vertices in a directed graph with directed paths between every ordered pair. Directed reachability in both directions is stricter than ordinary connectivity.
Vertex (graph theory): A vertex is a fundamental unit of a graph, connected to other vertices by edges. Components partition the graph’s vertices into maximal connected groups.
Breadth-first search: Breadth-first search explores a graph in successive layers of distance from a starting vertex. One search discovers all vertices reachable from its starting vertex.
Image segmentation: Image segmentation partitions an image into regions according to selected visual or semantic criteria. Connected components turn adjacent foreground pixels into distinct image regions.
Weakly connected component: A weakly connected component is a connected component of a directed graph after edge directions are ignored. It treats directed edges as undirected, unlike directed strong connectivity.
Edge (graph theory): An edge is a connection between two vertices in a graph. Edges determine which paths can connect vertices within a component.
Disjoint-set data structure: A disjoint-set data structure maintains a partition while supporting set merging and membership queries. Union operations combine endpoint sets as graph edges are processed.

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