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The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example (,) [,] = (,]. If R {\displaystyle \mathbb {R} } is viewed as a metric space , its open balls are the open bounded intervals ( c + r , c − r ) , and its closed balls ...
An indifference graph, formed from a set of points on the real line by connecting pairs of points whose distance is at most one. In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other. [1]
An interval graph is an undirected graph G formed from a family of intervals , =,,, … by creating one vertex v i for each interval S i, and connecting two vertices v i and v j by an edge whenever the corresponding two sets have a nonempty intersection.
An interval graph is a graph in which the nodes are 1-dimensional intervals (e.g. time intervals) and there is an edge between two intervals if and only if they intersect. An independent set in an interval graph is just a set of non-overlapping intervals.
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
It is interesting about interval chromatic number that it is easily computable. Indeed, by a simple greedy algorithm one can efficiently find an optimal partition of the vertex set of H into X < (H) independent intervals. This is in sharp contrast with the fact that even the approximation of the usual chromatic number of graph is an NP hard task.
Intermediate value theorem: Let be a continuous function defined on [,] and let be a number with () < < ().Then there exists some between and such that () =.. In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between () and () at some point within the interval.
The edges of the graph are d-tuples of intervals, one interval in every real line. [1] The simplest case is d = 1. The vertex set of a 1-interval hypergraph is the set of real numbers; each edge in such a hypergraph is an interval of the real line. For example, the set { [−2, −1], [0, 5], [3, 7] } defines a 1-interval