k-connected graph

Connectivity of graphs, when it isn’t specified which flavor is intended, usually refers to vertex connectivity, unless it is clear from the context that it refers to edge connectivity.

The (vertex) connectivity κ(G) is the minimum number of vertices (aka nodes) you have to remove to either make the graph no longer connectedPlanetmathPlanetmathPlanetmath, or reduce it to a single vertex (node). G is said to be k-(vertex)-connected for any kκ(G)𝖨𝖭. Note that “removing a vertex” in graph theoryMathworldPlanetmath also involves removing all the edges incidentPlanetmathPlanetmathPlanetmath to that vertex.

The edge connectivity κ(G) of a graph G is more straightforward, it is just the minimum number of edges you have to remove to make the graph no longer connected. G is said to be k-edge-connected for any kκ(G)𝖨𝖭. And note “removing an edge” is simply that; it does not entail removing any vertices.

  • If κ(G)=0 also κ(G)=0 and vice versa; such graphs are called disconnected and consist of several connected componentsMathworldPlanetmathPlanetmath. If κ and κ are nonzero (the graph is 1-connected) it is a connected graph (a single connected component)

  • If κ(G)=1 the graph contains at least one bridge. If κ(G)>1 (the graph is 2-edge-connected) every edge is part of a cycle (circuitMathworldPlanetmath, closed path).

    If κ(G)=1 the graph contains at least one cutvertex. If κ(G)>1 (the graph is 2-vertex-connected) there is, for any pair of vertices, a cycle they both lie on.

  • For the complete graphsMathworldPlanetmath we have κ(Kn)=κ(Kn)=n-1.

  • For any graph G we have κ(G)κ(G)δ(G) where the latter is the minimum valencyPlanetmathPlanetmath in G (the valency of a vertex is the number of edges sprouting from it).

Everything on this page applies equally well to multigraphsMathworldPlanetmath and pseudographsMathworldPlanetmath.

For directed graphsMathworldPlanetmath there are two notions of connectivity (http://planetmath.org/ConnectedGraph) (“weak” if the underlying graph is connected, “strong” if you can get from everywhere to everywhere).

There are now pictures (http://planetmath.org/ExamplesOfKConnectedGraphs) to go with this entry.

Title k-connected graph
Canonical name kconnectedGraph
Date of creation 2013-03-22 13:10:58
Last modified on 2013-03-22 13:10:58
Owner marijke (8873)
Last modified by marijke (8873)
Numerical id 7
Author marijke (8873)
Entry type Definition
Classification msc 05C40
Related topic ConnectedGraph
Related topic ClosedPath
Defines k-connected k-vertex-connected k-edge-connected connectivity