k-Connected Graph

A graph  on more than two vertices is said to be -connected (or -vertex connected, or -point connected) if there does not exist a vertex cut of size  whose removal disconnects the graph, i.e., if the vertex connectivity . Therefore, a connected graph on more than one vertex is 1-connected and a biconnected graph on more than two vertices is 2-connected.

The singleton graph is "annoyingly inconsistent" (West 2000, p. 150) since it is connected (specifically, 1-connected), but by convention it is taken to have .

The wheel graph is the "basic 3-connected graph" (Tutte 1961; Skiena 1990, p. 179).

-connectedness graph checking is implemented in the Wolfram Language as KVertexConnectedGraphQ[g, k].

The following table gives the numbers of -connected graphs for -node graphs (counting the singleton graph  as 1-connected and the path graph  as 2-connected).

	OEIS	-connected graphs on 1, 2, ... nodes
1	A001349	1, 1, 2, 6, 21, 112, 853, 11117, 261080, ...
2	A002218	0, 1, 1, 3, 10, 56, 468, 7123, 194066, ...
3	A006290	0, 0, 0, 1, 3, 17, 136, 2388, 80890, ...
4	A086216	0, 0, 0, 0, 1, 4, 25, 384, 14480, ...
5	A086217	0, 0, 0, 0, 0, 1, 4, 39, 1051, 102630, 22331311, ...
6	A324240	0, 0, 0, 0, 0, 0, 1, 5, 59, 3211, 830896, ...
7	A324092	0, 0, 0, 0, 0, 0, 0, 1, 5, 87, 9940, 7532629, ...

REFERENCES

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 45, 1994.

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.

Sloane, N. J. A. Sequences A000719/M1452, A001349/M1657, A002218/M2873, A006290/M3039, A052442, A052443, A052444, A052445, A086216, A086217, A324240, and A324092 in "The On-Line Encyclopedia of Integer Sequences."Tutte, W. T. "A Theory of 3-Connected Graphs." Indag. Math. 23, 441-455, 1961.

West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000.