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In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points ), together with a set of unordered pairs of these ...
Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs M i = μ(M i−1), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M 2 = K 2 with two vertices connected by an edge, the cycle graph M 3 = C 5 , and the Grötzsch graph M 4 with 11 vertices and 20 edges.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
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 ...
The vertex set of an n-vertex graph may be identified with the integers from 1 to n, and using such an identification a canonical form of a graph may also be described as a permutation of its vertices. Canonical forms of a graph are also called canonical labelings, [4] and graph canonization is also sometimes known as graph canonicalization.
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 .
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 ...
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.