A327127 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices where k is the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph.

1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 2, 0, 1, 13, 11, 7, 2, 0, 1

Offset

0,7

Comments

A graph with one vertex and no edges is considered to be connected. Except for complete graphs, this is the same as vertex-connectivity (A259862).

There are two ways to define (vertex) connectivity: the minimum size of a vertex cut, and the minimum of the maximum number of internally disjoint paths between two distinct vertices. For non-complete graphs they coincide, which is tremendously useful. For complete graphs with at least 2 vertices, there are no cuts but the second method still works so it is customary to use it to justify the connectivity of K_n being n-1. - Brendan McKay, Aug 28 2019.

Links

Table of n, a(n) for n=0..20.

Brendan McKay, confusion over k-connected graphs, posting to Sequence Fans Mailing List, Jul 08 2015.

Example

Triangle begins:
   1
   0  1
   1  0  1
   2  1  0  1
   5  3  2  0  1
  13 11  7  2  0  1

Crossrefs

Row sums are A000088.

Column k = 0 is A000719, if we assume A000719(0) = 1.

Column k = 1 is A052442, if we assume A052442(1) = 1 and A052442(2) = 0.

The labeled version is A327125.

A more standard version (zeros removed) is A259862.

Cf. A052443, A322389, A326786, A327082, A327098, A327100, A327113, A327126, A327128, A327197.

Sequence in context: A291883 A361957 A239145 * A151824 A275514 A180782

Adjacent sequences: A327124 A327125 A327126 * A327128 A327129 A327130

Keyword

nonn,more,tabl

Author

Gus Wiseman, Aug 25 2019

Status

approved