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.
A First Look At Graph Theory By John Clark Pdf 22
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 image analysis. 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 v \displaystyle v 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 v \displaystyle v .[1] Every graph is the disjoint union of its components.[2] Additional examples include the following special cases:
A graph can be interpreted as a topological space in multiple ways, for instance by placing its vertices as points in general position in three-dimensional Euclidean space and representing its edges as line segments between those points.[13] The components of a graph can be generalized through these interpretations as the topological connected components of the corresponding space; these are equivalence classes of points that cannot be separated by pairs of disjoint closed sets. Just as the number of connected components of a topological space is an important topological invariant, the zeroth Betti number, the number of components of a graph is an important graph invariant, and in topological graph theory it can be interpreted as the zeroth Betti number of the graph.[3]
It is straightforward to compute the components of a finite graph in linear time (in terms of the numbers of the vertices and edges of the graph) using either breadth-first search or depth-first search. In either case, a search that begins at some particular vertex v \displaystyle v will find the entire component containing v \displaystyle v (and no more) before returning. All components of a graph can be found by looping through its vertices, starting a new breadth-first or depth-first search whenever the loop reaches a vertex that has not already been included in a previously found component. Hopcroft & Tarjan (1973) describe essentially this algorithm, and state that it was already "well known".[19]
Connected-component labeling, a basic technique in computer image analysis, involves the construction of a graph from the image and component analysis on the graph.The vertices are the subset of the pixels of the image, chosen as being of interest or as likely to be part of depicted objects. Edges connect adjacent pixels, with adjacency defined either orthogonally according to the Von Neumann neighborhood, or both orthogonally and diagonally according to the Moore neighborhood. Identifying the connected components of this graph allows additional processing to find more structure in those parts of the image or identify what kind of object is depicted. Researchers have developed component-finding algorithms specialized for this type of graph, allowing it to be processed in pixel order rather than in the more scattered order that would be generated by breadth-first or depth-first searching. This can be useful in situations where sequential access to the pixels is more efficient than random access, either because the image is represented in a hierarchical way that does not permit fast random access or because sequential access produces better memory access patterns.[20]
Components of graphs have been used in computational complexity theory to study the power of Turing machines that have a working memory limited to a logarithmic number of bits, with the much larger input accessible only through read access rather than being modifiable. The problems that can be solved by machines limited in this way define the complexity class L. It was unclear for many years whether connected components could be found in this model, when formalized as a decision problem of testing whether two vertices belong to the same component, and in 1982 a related complexity class, SL, was defined to include this connectivity problem and any other problem equivalent to it under logarithmic-space reductions.[25] It was finally proven in 2008 that this connectivity problem can be solved in logarithmic space, and therefore that SL = L.[26]
For different models including the random subgraphs of grid graphs, the connected components are described by percolation theory. A key question in this theory is the existence of a percolation threshold, a critical probability above which a giant component (or infinite component) exists and below which it does not.[33] 2ff7e9595c
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