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Minor testing (checking whether an input graph contains an input graph as a minor); the same holds with topological minors; Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. [2] (The minimum spanning tree for an entire graph is solvable in polynomial time.) Modularity maximization [5]
Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)
The Mandelbrot set, one of the most famous examples of mathematical visualization. Mathematical phenomena can be understood and explored via visualization. Classically, this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century).
A face with a point hole is considered a monogonal hole, adding one vertex, and one edge, and can attached to a degenerate monogonal hosohedron hole, like a cylinder hole with zero radius. A face with a degenerate digon hole adds 2 vertices and 2 coinciding edges, where the two edges attach to two coplanar faces, as a dihedron hole.
The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. [2] [3] [4] The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities.
Such shapes include polygons, which may be concave, shapes with holes, and collections of such shapes (i.e. the regions need not be connected). The corresponding problem that the theorem solves is known as the fold-and-cut problem, which asks what shapes can be obtained by the so-called fold-and-cut method.
Sightings of hole-punch clouds inundated the Facebook page for the National Weather Service in Peachtree City, Georgia, on Monday with more than a dozen photos sent in from across northern Georgia.
It originates from the following real-world problem: "In an art gallery, what is the minimum number of guards who together can observe the whole gallery?" In the geometric version of the problem, the layout of the art gallery is represented by a simple polygon and each guard is represented by a point in the polygon.