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All ordinary linear transformations are included in the set of affine transformations, and can be described as a simplified form of affine transformations. Therefore, any linear transformation can also be represented by a general transformation matrix.
The form of YΔ- and ΔY-transformations used to define the Petersen family is as follows: . If a graph G contains a vertex v with exactly three neighbors, then the YΔ-transform of G at v is the graph formed by removing v from G and adding edges between each pair of its three neighbors.
Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One way to understand this is to say we have a vector at an angle 30° from the x axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°.
ΔY- and YΔ-transformations are a tool both in pure graph theory as well as applications. Both operations preserve a number of natural topological properties of graphs. . For example, applying a YΔ-transformation to a 3-vertex of a planar graph, or a ΔY-transformation to a triangular face of a planar graph, results again in a planar graph.
The vertex cover problem parameterized by the feedback vertex number of the input graph has a polynomial kernelization: [9] There is a polynomial-time algorithm that, given a graph whose feedback vertex number is , outputs a graph ′ on () vertices such that a minimum vertex cover in ′ can be transformed into a minimum vertex cover for in ...
A proper subset of the vertices of a polytope forms the vertex set of a face of the polytope, if and only if the complementary set of vectors of the Gale transform has a convex hull that contains the origin in its relative interior. Equivalently, the subset of vertices forms a face if and only if its affine span does not intersect the convex ...
In mathematics, a transformation, transform, or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [ 2 ] [ 3 ] [ 4 ] Examples include linear transformations of vector spaces and geometric transformations , which include projective transformations , affine transformations , and ...
The blue vertices form a minimum vertex cover, and the blue vertices in the gray oval correspond to a satisfying truth assignment for the original formula. In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently efficient reduction from one ...