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Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
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 ...
The precise analysis of the performance of a disjoint-set forest is somewhat intricate. However, there is a much simpler analysis that proves that the amortized time for any m Find or Union operations on a disjoint-set forest containing n objects is O(m log * n), where log * denotes the iterated logarithm. [12] [13] [14] [15]
A complete bipartite graph with m = 5 and n = 3 The Heawood graph is bipartite.. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in .
A domatic partition. In graph theory, a domatic partition of a graph = (,) is a partition of into disjoint sets , ,..., such that each V i is a dominating set for G.The figure on the right shows a domatic partition of a graph; here the dominating set consists of the yellow vertices, consists of the green vertices, and consists of the blue 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 ...
By deleting just one edge of the spanning tree, the vertices are partitioned into two disjoint sets. The fundamental cutset is defined as the set of edges that must be removed from the graph G to accomplish the same partition. Thus, each spanning tree defines a set of V − 1 fundamental cutsets, one for each edge of the spanning tree. [6]
The optimization version of the problem, maximum set packing, asks for the maximum number of pairwise disjoint sets in the list. It is a maximization problem that can be formulated naturally as an integer linear program , belonging to the class of packing problems .