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A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction). A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, [5] a directed graph is an ordered pair = (,) comprising:
Given a graph , another graph is -saturated if does not contain a (not necessarily induced) copy of , but adding any edge to it does. The function s a t ( n , H ) {\displaystyle sat(n,H)} is the minimum number of edges an H {\displaystyle H} -saturated graph G {\displaystyle G} on n {\displaystyle n} vertices can have.
The function depends on three parameters, the input x, the "left edge" and the "right edge", with the left edge being assumed smaller than the right edge. The function receives a real number x as an argument and returns 0 if x is less than or equal to the left edge, 1 if x is greater than or equal to the right edge, and smoothly interpolates ...
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.
In an undirected simple graph, an edge may be represented as the set of its vertices, and in a directed simple graph it may be represented as an ordered pair of its vertices. An edge that connects vertices x and y is sometimes written xy. edge cut A set of edge s whose removal disconnects the graph. A one-edge cut is called a bridge, isthmus ...
In the mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn.
A minimum edge covering is an edge covering of smallest possible size. The edge covering number ρ(G) is the size of a minimum edge covering. The following figure shows examples of minimum edge coverings (again, the set C is marked with red). Note that the figure on the right is not only an edge cover but also a matching.