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.
The components of a graph can be constructed in linear time, and a special case of the problem, connected-component labeling, is a basic technique in . Dynamic connectivity algorithms maintain components as edges are inserted or deleted in a graph, in low time per change. In computational complexity theory, connected components have been used to study algorithms with limited space complexity, and sublinear time algorithms can accurately estimate the number of components.
A component of a given undirected graph may be defined as a connected subgraph that is not part of any larger connected subgraph. For instance, the graph shown in the first illustration has three components. Every vertex of a graph belongs to one of the graph's components, which may be found as the induced subgraph of the set of vertices reachable from . Every graph is the disjoint union of its components. Additional examples include the following special cases:
In an empty graph, each vertex forms a component with one vertex and zero edges. More generally, a component of this type is formed for every isolated vertex in any graph.
In a connected graph, there is exactly one component: the whole graph.
In a forest, every component is a tree.
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Borůvka's algorithm is a greedy algorithm for finding a minimum spanning tree in a graph, or a minimum spanning forest in the case of a graph that is not connected. It was first published in 1926 by Otakar Borůvka as a method of constructing an efficient electricity network for Moravia. The algorithm was rediscovered by Choquet in 1938; again by Florek, Łukasiewicz, Perkal, Steinhaus, and Zubrzycki in 1951; and again by Georges Sollin in 1965. This algorithm is frequently called Sollin's algorithm, especially in the parallel computing literature.
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.
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