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A cut C = (S, T) is a partition of V of a graph G = (V, E) into two subsets S and T. The cut-set of a cut C = (S, T) is the set {(u, v) ∈ E | u ∈ S, v ∈ T} of edges that have one endpoint in S and the other endpoint in T. If s and t are specified vertices of the graph G, then an s – t cut is a cut in which s belongs to the set S and t ...
Multi-colored vertices are cut vertices, and thus belong to multiple biconnected components. In graph theory, a biconnected component or block (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph.
A vertex (plural vertices) is (together with edges) one of the two basic units out of which graphs are constructed. Vertices of graphs are often considered to be atomic objects, with no internal structure. vertex cut separating set A set of vertices whose removal disconnects the graph. A one-vertex cut is called an articulation point or cut vertex.
A vertex cut or separating set of a connected graph G is a set of vertices whose removal renders G disconnected. The vertex connectivity κ(G) (where G is not a complete graph) is the size of a smallest vertex cut. A graph is called k-vertex-connected or k-connected if its vertex connectivity is k or greater.
A cut (,) in an undirected graph = (,) is a partition of the vertices into two non-empty, disjoint sets =.The cutset of a cut consists of the edges {:,} between the two parts. . The size (or weight) of a cut in an unweighted graph is the cardinality of the cutset, i.e., the number of edges between the two parts
The dotted line in red represents a cut with three crossing edges. The dashed line in green represents one of the minimum cuts of this graph, crossing only two edges. [1] In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric.
An example of a maximum cut. In a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of edges between S and T is as large as possible. Finding such a cut is known as the max-cut problem.
The vertex-connectivity statement of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two nonadjacent vertices. Then the size of the minimum vertex cut for x and y (the minimum number of vertices, distinct from x and y, whose removal disconnects x and y) is equal to the maximum number of pairwise internally disjoint paths from x to y.