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The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).
For a parametric curve this is an easy task: One just computes the points of a sequence of parametric values. For an implicit curve one has to solve two subproblems: determination of a first curve point to a given starting point in the vicinity of the curve, determination of a curve point starting from a known curve point.
By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval [−2, 2] and the image is [−1, 1]; for the second one, the domain is [−2, ∞) and the image is [1, ∞); for the last one, the domain is (−∞, 2] and the image is (−∞, −1]. As the three graphs together form a ...
Forbidden graph characterizations may be used in algorithms for testing whether a graph belongs to a given family. In many cases, it is possible to test in polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set.
In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges (,) an -vertex graph can have such that it does not have a subgraph isomorphic to .
Defining g −1 as the inverse of g is an implicit definition. For some functions g, g −1 (y) can be written out explicitly as a closed-form expression — for instance, if g(x) = 2x − 1, then g −1 (y) = 1 / 2 (y + 1). However, this is often not possible, or only by introducing a new notation (as in the product log example below).
In the graph, moving one unit to the right (increasing x by 1) moves the y-value up by a: that is, (+) = +. Negative slope a indicates a decrease in y for each increase in x . For example, the linear function y = − 2 x + 4 {\displaystyle y=-2x+4} has slope a = − 2 {\displaystyle a=-2} , y -intercept point ( 0 , b ) = ( 0 , 4 ...
A graph is said to be k-factor-critical if every subset of n − k vertices has a perfect matching. Under this definition, a hypomatchable graph is 1-factor-critical. [13] Even more generally, a graph is (a,b)-factor-critical if every subset of n − k vertices has an r-factor, that is, it is the vertex set of an r-regular subgraph of the given ...