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The graph shown here appears as a subgraph of an undirected graph if and only if models the sentence ,,,... In the first-order logic of graphs, a graph property is expressed as a quantified logical sentence whose variables represent graph vertices, with predicates for equality and adjacency testing.
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 ...
The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The data structure used depends on both the graph structure and the algorithm used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete ...
For instance, in the logic of graphs, testing whether a graph is disconnected belongs to monadic NP, as the test can be represented by a formula that describes the existence of a proper subset of vertices with no edges connecting them to the rest of the graph; however, the complementary problem, testing whether a graph is connected, does not ...
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).
An ordered pair of vertices, such as an edge in a directed graph. An arrow (x, y) has a tail x, a head y, and a direction from x to y; y is said to be the direct successor to x and x the direct predecessor to y. The arrow (y, x) is the inverted arrow of the arrow (x, y). articulation point A vertex in a connected graph whose removal would ...
Graph structure theorem. For any graph H, there exists a positive integer k such that every H-free graph can be obtained as follows: We start with a list of graphs, where each graph in the list is embedded on a surface on which H does not embed; to each embedded graph in the list, we add at most k vortices, where each vortex has depth at most k
Given a graph G(V,E) and a subset T⊆V, define cutset(T) as the set of edges that connect T with V\T. The cutset structure is a data structure that, without keeping the entire graph in memory, can quickly find an edge in the cutset, if such an edge exists. [7] Start by giving a number to each vertex.