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."