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The sparsest cut problem is to bipartition the vertices so as to minimize the ratio of the number of edges across the cut divided by the number of vertices in the smaller half of the partition. This objective function favors solutions that are both sparse (few edges crossing the cut) and balanced (close to a bisection).
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.
A cut whose cut-set has minimum total weight, possibly restricted to cuts that separate a designated pair of vertices; they are characterized by the max-flow min-cut theorem. minor A graph H is a minor of another graph G if H can be obtained by deleting edges or vertices from G and contracting edges in G.
A graph with 16 vertices and six bridges (highlighted in red) An undirected connected graph with no bridge edges. In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. [1] Equivalently, an edge is a bridge if and only if it is not contained in any cycle.
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.
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.
The Riemann Hypothesis. Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize ...
A drawing of 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. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links or lines).