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A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A map graph is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they ...
The concept of a combinatorial map was introduced informally by J. Edmonds for polyhedral surfaces [2] which are planar graphs.It was given its first definite formal expression under the name "Constellations" by A. Jacques [3] [4] but the concept was already extensively used under the name "rotation" by Gerhard Ringel [5] and J.W.T. Youngs in their famous solution of the Heawood map-coloring ...
This auxiliary graph is 1-planar, from which it follows that Ringel's vertex-face coloring problem may also be solved with six colors. [2] The graph K 6 cannot be formed as an auxiliary graph in this way, but nevertheless the vertex-face coloring problem also sometimes requires six colors; for instance, if the planar graph to be colored is a ...
Let H = {h 1, h 2, ..., h k} be the convex hull of P; then the farthest-point Voronoi diagram is a subdivision of the plane into k cells, one for each point in H, with the property that a point q lies in the cell corresponding to a site h i if and only if d(q, h i) > d(q, p j) for each p j ∈ S with h i ≠ p j, where d(p, q) is the Euclidean ...
In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve . Phase portraits are an invaluable tool in studying dynamical systems.
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc. (any pair of variables).
The Tutte polynomial factors into connected components. If is the union of disjoint graphs and ′ then = ′ If is planar and denotes its dual graph then (,) = (,)Especially, the chromatic polynomial of a planar graph is the flow polynomial of its dual.
A smooth plane curve is a curve in a real Euclidean plane and is a one-dimensional smooth manifold.This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function.