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The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. [3] As such, it is unlikely that there exists an efficient algorithm for finding a maximum independent set of a graph. Every maximum independent set also is maximal, but the converse implication does not necessarily hold.
In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function : [,] [,], that is important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeable random graph models.
This is in general not testable from the data, but if the data are known to be dependent (e.g. paired by test design), a dependent test has to be applied. For partially paired data, the classical independent t -tests may give invalid results as the test statistic might not follow a t distribution, while the dependent t -test is sub-optimal as ...
However, not all functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.
(This is the Cesàro limit of the indicator functions. In cases where the Cesàro limit does not exist this function can actually be defined as the Banach limit of the indicator functions, which is an extension of this limit. This latter limit always exists for sums of indicator functions, so that the empirical distribution is always well-defined.)
In mathematics, the nth-term test for divergence [1] is a simple test for the divergence of an infinite series: If lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to \infty }a_{n}\neq 0} or if the limit does not exist, then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} diverges.
The top two P 3 graphs are maximal independent sets while the bottom two are independent sets, but not maximal. The maximum independent set is represented by the top left. A graph may have many MISs of widely varying sizes; [a] the largest, or possibly several equally large, MISs of a graph is called a maximum independent set.
The clique problem and the independent set problem are complementary: a clique in G is an independent set in the complement graph of G and vice versa. [20] Therefore, many computational results may be applied equally well to either problem, and some research papers do not clearly distinguish between the two problems.