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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 ...
Let S be an (a,b)-separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a minimal (a,b)-separator if no proper subset of S separates a and b. More generally, S is called a minimal separator if it is a minimal separator for some pair (a,b) of nonadjacent vertices.
The complete graph on n vertices is denoted by K n.Some sources claim that the letter K in this notation stands for the German word komplett, [4] but the German name for a complete graph, vollständiger Graph, does not contain the letter K, and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
Multiple edges, not allowed under the definition above, are two or more edges with both the same tail and the same head. In one more general sense of the term allowing multiple edges, [5] a directed graph is an ordered triple = (,,) comprising: , a set of vertices (also called nodes or points);
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.
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.