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  2. Intersection number (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Intersection_number_(graph...

    In the mathematical field of graph theory, the intersection number of a graph = (,) is the smallest number of elements in a representation of as an intersection graph of finite sets. In such a representation, each vertex is represented as a set, and two vertices are connected by an edge whenever their sets have a common element.

  3. Scheinerman's conjecture - Wikipedia

    en.wikipedia.org/wiki/Scheinerman's_conjecture

    In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis , following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane (Ehrlich, Even & Tarjan 1976).

  4. Intersection number - Wikipedia

    en.wikipedia.org/wiki/Intersection_number

    Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:

  5. P versus NP problem - Wikipedia

    en.wikipedia.org/wiki/P_versus_NP_problem

    In other words, any problem in EXPTIME is solvable by a deterministic Turing machine in O(2 p(n)) time, where p(n) is a polynomial function of n. A decision problem is EXPTIME-complete if it is in EXPTIME, and every problem in EXPTIME has a polynomial-time many-one reduction to it. A number of problems are known to be EXPTIME-complete.

  6. De Morgan's laws - Wikipedia

    en.wikipedia.org/wiki/De_Morgan's_laws

    De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.

  7. Buffon's needle problem - Wikipedia

    en.wikipedia.org/wiki/Buffon's_needle_problem

    For simplicity in the algebraic formulation ahead, let a = b = t = 2l such that the original result in Buffon's problem is P(A) = P(B) = ⁠ 1 / π ⁠. Furthermore, let N = 100 drops. Now let us examine P(AB) for Laplace's result, that is, the probability the needle intersects both a horizontal and a vertical line. We know that

  8. Graph operations - Wikipedia

    en.wikipedia.org/wiki/Graph_operations

    graph intersection: G 1 ∩ G 2 = (V 1 ∩ V 2, E 1 ∩ E 2); [1] graph join: . Graph with all the edges that connect the vertices of the first graph with the vertices of the second graph. It is a commutative operation (for unlabelled graphs); [2] graph products based on the cartesian product of the vertex sets: cartesian graph product: it is a ...

  9. Intersection graph - Wikipedia

    en.wikipedia.org/wiki/Intersection_graph

    The line graph of a graph G is defined as the intersection graph of the edges of G, where we represent each edge as the set of its two endpoints. A string graph is the intersection graph of curves on a plane. A graph has boxicity k if it is the intersection graph of multidimensional boxes of dimension k, but not of any smaller dimension.