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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).
If a graph is both a split graph and an interval graph, then its complement is both a split graph and a comparability graph, and vice versa. The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs. [7] The split cographs are exactly the threshold ...
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 part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently, it is a graph that can be colored with k colors, so that no two endpoints of an edge have the same color. When k = 2 these are the bipartite graphs, and when k = 3 they are called the ...
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,
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 ...
A clique cover of a graph G may be seen as a graph coloring of the complement graph of G, the graph on the same vertex set that has edges between non-adjacent vertices of G. Like clique covers, graph colorings are partitions of the set of vertices, but into subsets with no adjacencies (independent sets) rather than cliques.
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.