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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 ...
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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 one variation of monadic second-order graph logic known as MSO 1, the graph is described by a set of vertices and a binary adjacency relation (.,.), and the restriction to monadic logic means that the graph property in question may be defined in terms of sets of vertices of the given graph, but not in terms of sets of edges, or sets of tuples of vertices.
A chain graph is a graph which may have both directed and undirected edges, but without any directed cycles (i.e. if we start at any vertex and move along the graph respecting the directions of any arrows, we cannot return to the vertex we started from if we have passed an arrow). Both directed acyclic graphs and undirected graphs are special ...
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 ...
An example for an undirected Graph with a vertex r and its corresponding level structure For the concept in algebraic geometry, see level structure (algebraic geometry) In the mathematical subfield of graph theory a level structure of a rooted graph is a partition of the vertices into subsets that have the same distance from a given root vertex.
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
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