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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 ...
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.
Similarly, if v is an articulation point of a graph that is not 2-vertex-connected, then the graph has multiple splits in which v and some but not all of the components formed by its deletion are on one side, and the remaining components are on the other side. In these examples, the cut-set of the split forms a star.
In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff. [1] Nested dissection consists of the following steps:
Partition refinement also forms a key step in lexicographic breadth-first search, a graph search algorithm with applications in the recognition of chordal graphs and several other important classes of graphs. Again, the disjoint set elements are vertices and the set X represents sets of neighbors, so the algorithm takes linear time. [8] [9]
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set,
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 ...
In graph theory, a discipline within mathematics, the frequency partition of a graph (simple graph) is a partition of its vertices grouped by their degree. For example, the degree sequence of the left-hand graph below is (3, 3, 3, 2, 2, 1) and its frequency partition is 6 = 3 + 2 + 1. This indicates that it has 3 vertices with some degree, 2 ...