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1-connectedness is equivalent to connectedness for graphs of at least two vertices. The complete graph on n vertices has edge-connectivity equal to n − 1. Every other simple graph on n vertices has strictly smaller edge-connectivity. In a tree, the local edge-connectivity between any two distinct vertices is 1.
Equivalently, the connectivity of a graph is the greatest integer k for which the graph is k-connected. While terminology varies, noun forms of connectedness-related properties often include the term connectivity. Thus, when discussing simply connected topological spaces, it is far more common to speak of simple connectivity than simple ...
The vertex-connectivity of an input graph G can be computed in polynomial time in the following way [4] consider all possible pairs (,) of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for (,) is the number of pairwise vertex-independent paths between them, encode the input by doubling each vertex as an edge to reduce to a computation of the ...
Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order.
Several algorithms based on depth-first search compute strongly connected components in linear time.. Kosaraju's algorithm uses two passes of depth-first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth-first search tests vertices for having been visited already and recursively explores them if not.
In applied mathematics, lambda-connectedness (or λ-connectedness) deals with partial connectivity for a discrete space. Assume that a function on a discrete space (usually a graph ) is given. A degree of connectivity (connectedness) will be defined to measure the connectedness of the space with respect to the function.
Niel Ritchie is the executive director of the League of Rural Voters. This article originally appeared on Des Moines Register: Net neutrality vs. rural connectivity: FCC making wrong choice
The edge connectivity of is the maximum value k such that G is k-edge-connected. The smallest set X whose removal disconnects G is a minimum cut in G . The edge connectivity version of Menger's theorem provides an alternative and equivalent characterization, in terms of edge-disjoint paths in the graph.