Search results
Results from the WOW.Com Content Network
The input to the algorithm is an undirected graph G = (V, E) with vertex set V, edge set E, and (optionally) numerical weights on the edges in E.The goal of the algorithm is to partition V into two disjoint subsets A and B of equal (or nearly equal) size, in a way that minimizes the sum T of the weights of the subset of edges that cross from A to B.
The example in Figure 3 illustrates 2 instances of the same graph such that in (a) modularity (Q) is the partitioning metric and in (b), ratio-cut is the partitioning metric. Figure 3: Weighted graph G may be partitioned to maximize Q in (a) or to minimize the ratio-cut in (b).
Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms of the number of edges between parts ...
In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.
These graphs are described by notation with a capital letter K subscripted by a sequence of the sizes of each set in the partition. For instance, K 2,2,2 is the complete tripartite graph of a regular octahedron , which can be partitioned into three independent sets each consisting of two opposite vertices.
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 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 ...
As another component of the Dulmage–Mendelsohn decomposition, Dulmage and Mendelsohn defined the core of a graph to be the union of its maximum matchings. [5] However, this concept should be distinguished from the core in the sense of graph homomorphisms, and from the k-core formed by the removal of low-degree vertices.