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Finding a partition that simplifies graph analysis is a hard problem, but one that has applications to scientific computing, VLSI circuit design, and task scheduling in multiprocessor computers, among others. [1] Recently, the graph partition problem has gained importance due to its application for clustering and detection of cliques in social ...
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.
A classical approach to solve the Hypergraph bipartitioning problem is an iterative heuristic by Charles Fiduccia and Robert Mattheyses. [1] This heuristic is commonly called the FM algorithm. Introduction
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.
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.
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.
In some applications of partition refinement, such as lexicographic breadth-first search, the data structure maintains as well an ordering on the sets in the partition. Partition refinement forms a key component of several efficient algorithms on graphs and finite automata , including DFA minimization , the Coffman–Graham algorithm for ...
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.