${k}$ -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 $\kappa(G)$ is the minimum number of vertices (aka nodes) you have to remove to either make the graph no longer connected, or reduce it to a single vertex (node). $G$ is said to be $k$ -(vertex)-connected for any $k\leqslant\kappa(G)\in{\mathchoice{{\sf I\kern-1.2ptN}}{{\sf I\kern-1.2ptN}}{{% \scriptstyle\sf I\kern-1.2ptN}}{{\scriptscriptstyle\sf I\kern-1.2ptN}}}$ . Note that “removing a vertex” in graph theory also involves removing all the edges incident to that vertex.

The edge connectivity $\kappa^{\prime}(G)$ 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 $k\leqslant\kappa^{\prime}(G)\in{\mathchoice{{\sf I\kern-1.2ptN}}{{\sf I\kern-1% .2ptN}}{{\scriptstyle\sf I\kern-1.2ptN}}{{\scriptscriptstyle\sf I\kern-1.2ptN}}}$ . And note “removing an edge” is simply that; it does not entail removing any vertices.

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If $\kappa^{\prime}(G)=0$ also $\kappa(G)=0$ and vice versa; such graphs are called disconnected and consist of several connected components. If $\kappa$ and $\kappa^{\prime}$ are nonzero (the graph is 1-connected) it is a connected graph (a single connected component)
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If $\kappa^{\prime}(G)=1$ the graph contains at least one bridge. If $\kappa^{\prime}(G)>1$ (the graph is 2-edge-connected) every edge is part of a cycle (circuit, closed path).

If $\kappa(G)=1$ the graph contains at least one cutvertex. If $\kappa(G)>1$ (the graph is 2-vertex-connected) there is, for any pair of vertices, a cycle they both lie on.
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For the complete graphs we have $\kappa(K_{n})=\kappa^{\prime}(K_{n})=n-1$ .
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For any graph $G$ we have $\kappa(G)\leqslant\kappa^{\prime}(G)\leqslant\delta(G)$ where the latter is the minimum valency in $G$ (the valency of a vertex is the number of edges sprouting from it).

Everything on this page applies equally well to multigraphs and pseudographs.

For directed graphs 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

k-connected graph

${k}$ -connected graph