A minimal cover is a cover for which removal of any single member destroys the covering property. For example, of the five covers
of , namely , , , , and , only and are minimal covers.
Similarly, the minimal covers of are given by , , , , , , , and . The numbers of minimal covers of members for , 2, ..., are 1, 2, 8, 49, 462, 6424, 129425, ... (OEIS A046165).
Royle (2000) proved that there is a one-one correspondence between the split graphs on
vertices and the minimal covers of a set of size .
Let
be the number of minimal covers of with members. Then
Hearne, T. and Wagner, C. "Minimal Covers of Finite Sets." Disc. Math.5, 247-251, 1973.Macula, A. J.
"Covers of a Finite Set." Math. Mag.67, 141-144, 1994.Macula,
A. J. "Lewis Carroll and the Enumeration of Minimal Covers." Math.
Mag.68, 269-274, 1995.Royle, G. F. "Counting Set
Covers and Split Graphs." J. Integer Seq.3, Article 00.2.6, 2000.
https://cs.uwaterloo.ca/journals/JIS/VOL3/ROYLE/royle.html.Sloane,
N. J. A. Sequences A000392, A003468,
A016111, A035348,
A046165, A046166,
A046167, A046168,
and A057668 in "The On-Line Encyclopedia
of Integer Sequences."
, namely 
, 
, 
, 
, and 
, only 
and 
are minimal covers.
are given by 
, 
, 
, 
, 
, 
, 
, and 
. The numbers of minimal covers of 
members for 
, 2, ..., are 1, 2, 8, 49, 462, 6424, 129425, ... (OEIS A046165).
vertices and the minimal covers of a set of size 
.
be the number of minimal covers of 
with 
members. Then
is a binomial coefficient, 
is a Stirling
number of the second kind, and
and 
.
The table below gives the a triangle of 
(OEIS A035348).
