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This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed.. In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more ...
A graph with 6 vertices and 7 edges. In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
Connectedness is a basic measure in many areas of mathematical science and social sciences. In graph theory, two vertices are said to be connected if there is a path between them. In topology, two points are connected if there is a continuous function that could move from one point to another continuously. In management science, for example, in ...
For example, a graph is said to be connected if each pair of vertices in the graph is joined by a path. This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the context of graph theory. Graph theory also offers a context-free measure of connectedness, called the clustering coefficient.
A graph with connectivity 4. In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected.
A graph with three components. 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 ...
In graph theory, a connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed. The edge-connectivity of a graph is the largest k for which the graph is k-edge-connected. Edge connectivity and the enumeration of k-edge-connected graphs was studied by Camille Jordan in 1869. [1]
In computing and graph theory, a dynamic connectivity structure is a data structure that dynamically maintains information about the connected components of a graph. The set V of vertices of the graph is fixed, but the set E of edges can change. The three cases, in order of difficulty, are: