Graph Theory Connectivity
The edge connectivity of a disconnected graph is 0, while that of a connected graph with a graph bridge is 1. Let kappa(G) be the vertex connectivity of a graph G. Both the edge connectivity and the vertex connectivity are characteristics a relation between the edge connectivity $\lambda$, the vertex connectivity $\ kappa$. has 2. vertices and 2. −1. edges. It is -regular. 4d-hypercube The edge-connectivity of a graph is the What is the relation between the.
It is strongly connected, diconnected, or simply strong if it contains a directed path from u to v and a directed path from v to u for every pair of vertices u, v.
The strong components are the maximal strongly connected subgraphs. A cutvertex cut, or separating set of a connected graph G is a set of vertices whose removal renders G disconnected. A graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater.
A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v.
A graph G which is connected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges. In the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, an edge cut of G is a set of edges whose removal renders the graph disconnected. Again, local edge-connectivity is symmetric.
A graph is called k-edge-connected if its edge connectivity is k or greater. A graph is said to be maximally connected if its connectivity equals its minimum degree.
Edge and Vertex Connectivity
A graph is said to be maximally edge-connected if its edge-connectivity equals its minimum degree. Menger's theorem One of the most important facts about connectivity in graphs is Menger's theoremwhich characterizes the connectivity and edge-connectivity of a graph in terms of the number of independent paths between vertices. If u and v are vertices of a graph G, then a collection of paths between u and v is called independent if no two of them share a vertex other than u and v themselves.
Similarly, the collection is edge-independent if no two paths in it share an edge. Computational aspects[ edit ] The problem of determining whether two vertices in a graph are connected can be solved efficiently using a search algorithmsuch as breadth-first search.
More generally, it is easy to determine computationally whether a graph is connected for example, by using a disjoint-set data structureor to count the number of connected components.
A simple algorithm might be written in pseudo-code as follows: Begin at any arbitrary node of the graph, G Proceed from that node using either depth-first or breadth-first search, counting all nodes reached. The vertex connectivity is always no smaller than the edge connectivity, since deleting one vertex incident on each edge in a cut set succeeds in disconnecting the graph.
Of course, smaller vertex subsets may be possible. The minimum vertex degree is an upper bound on both the edge and vertex connectivity, since deleting all its neighbors or the edges to all its neighbors disconnects the graph into one big and one single-vertex component.
Connectivity (graph theory)
Several connectivity problems prove to be of interest: Is the graph already disconnected? A simple depth-first or breadth-first search suffices to identify all components in linear time, as discussed in Section. For directed graphs, the issue is whether the graph is strongly connected, meaning there is a directed path between any pair of vertices. In a weakly connected graph, there may exist paths to nodes from which there is no way to return. What if I want to split the graph into equal-sized pieces?
For example, suppose we want to break a computer program spread across several files into two maintainable units. Construct a graph where the vertices are subroutines, with an edge between any two subroutines that interact, say by one calling the other. We seek to partition the routines into equal-sized sets so that the fewest pairs of interacting routines are spread across the set.
This is the graph partition problem, which is further discussed in Section. Although the problem is NP-complete, reasonable heuristics exist to solve it.
Is there one weak link in my graph? Any vertex that is such a weak point is called an articulation vertex. A bridge is the analogous concept for edges, meaning a single edge whose deletion disconnects the graph.
The simplest algorithms for identifying articulation vertices or bridges would try deleting vertices or edges one by one, and then using DFS or BFS to test whether the resulting graph is still connected. More complicated but linear-time algorithms exist for both problems, based on depth-first search.
Implementations are described below and in Section.