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Consider a graph G = (V, E), where V denotes the set of n vertices and E the set of edges. For a (k,v) balanced partition problem, the objective is to partition G into k components of at most size v · (n/k), while minimizing the capacity of the edges between separate components. [1]
A split graph may have more than one partition into a clique and an independent set; for instance, the path a–b–c is a split graph, the vertices of which can be partitioned in three different ways:
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 ...
In a cycle graph of length four, the partition of the vertices given by 2-coloring the cycle is a nontrivial split, but for cycles of any longer length there are no nontrivial splits. A bridge of a graph that is not 2-edge-connected corresponds to a split, with each side of the split formed by the vertices on one side of the bridge. The cut-set ...
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.
In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions. In a flow network , an s–t cut is a cut that requires the source and the sink to be in different subsets, and its cut-set only consists of edges going from the source's side to the ...
A graph factorization is a partition of the edges of the graph into factors; a k-factorization is a partition into k-factors. For instance a 1-factorization is an edge coloring with the additional property that each vertex is incident to an edge of each color. family A synonym for class. finite
An example for an undirected Graph with a vertex r and its corresponding level structure For the concept in algebraic geometry, see level structure (algebraic geometry) In the mathematical subfield of graph theory a level structure of a rooted graph is a partition of the vertices into subsets that have the same distance from a given root vertex.