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In mathematics and computer science the connectivity of graphs is one of the basic concepts of graph theory. It is closely related to the concept of path. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ...
Wikibooks has more about this subject: Wikiversity Riverside Graphics Lab Open Directory Project: Computer Science Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: Computer science | Academic disciplines ...
In mathematics and computer science, graph theory studies the properties of graphs. ...
The word path has a variety of meanings: a path is a route between two points. ...
In network theory we are often interested how many connections (edges) in the network are allowed to fail before two nodes (vertices) become disconnected. Menger's theorem and Fulkerson's theorem provide characterizations of this problem. Network theory is a branch of applied mathematics, with the same general subject matter as graph theory, namely the idea of a graph as a representation of a symmetric relation, and of a directed graph for a general binary relation. ...
This article just presents the basic definitions. ...
This article just presents the basic definitions. ...
In the mathematical discipline of graph theory and related areas Mengers theorem is a basic result about connectivity in finite undirected graphs. ...
By adding weights to the edges of the graph we are able to consider network flow problems which allow a finer discussion of connectivity. In graph theory, a network flow is an assignment of values to edges of a weighted directed graph (called a flow network in this case), such that: 1. ...
Checking if a graph is connected, or even its number of connected components, is a very easy problem that can be solved in deterministic logarithmic space (see SL). In computational complexity theory, SL (Symmetric Logspace) is the complexity class of problems log-space reducible to USTCON, which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same...
Connections
Given an undirected graph, two vertices u and v are called connected if there exists a path from u to v. Otherwise they are called disconnected. The graph is called connected graph if every pair of vertices in the graph is connected. This article just presents the basic definitions. ...
Cuts A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. A vertex cut for the whole graph is a set of vertices whose removal renders the graph disconnected. The vertex connectivity κ(G) for a graph G is the size of minimum vertex cut. A graph is called k-vertex-connected if its vertex connectivity is k or greater.
The same concept can be defined for edges.
An edge cut for two vertices u and v is a set of edges whose removal from the graph disconnects u and v. A edge cut for the whole graph is a set of edges whose removal renders the graph disconnected. The edge connectivity κ'(G) for a graph G is the size of the minimum edge cut. A graph is called k-edge-connected if its edge connectivity is k or greater.
Examples - The vertex and edge connectivities of a disconnected graph are both 0
- A connected graph is 1-connected by definition
- A complete graph is maximally connected; if it has n vertices, its edge connectivity is n
- A tree is minimally connected; its edge connectivity is 1
In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. ...
A tree with 6 vertices and 5 edges In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. ...
Properties - Connectedness is preserved by graph homomorphisms.
- If G is connected then the line graph L(G) is connected too.
- A graph is k-vertex-connected if and only if it contains k vertex independent paths between any two vertices.
- A graph is k-edge-connected if and only if it contains k edge independent paths between any two edges.
- Given a k-vertex-connected graph G then for every set of vertices U there exists a cycle in G containing U
- Given a connected graph the distance between two vertices u and v is equal to the number of pairwise disjunct edge cut sets which separate u and v.
In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. ...
In graph theory, the line graph L(G) of a graph G is a graph such that each vertex of L(G) represents an edge of G; and any two vertices of L(G) are adjacent if and only if their corresponding edges are incident, meaning they share a common...
One major problem that has plagued graph theory since its inception is the consistent lack of consistency in terminology. ...
In the mathematical subfield of graph theory we can define a notion of distance between two vertices in a graph. ...
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