OFFSET

0,4

COMMENTS

Covering means there are no isolated vertices.

A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also graphs with minimum vertex-degree 1.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50

EXAMPLE

The a(4) = 31 edge-sets:

{12,34} {12,13,14} {12,13,14,23}

{13,24} {12,13,24} {12,13,14,24}

{14,23} {12,13,34} {12,13,14,34}

{12,14,23} {12,13,23,24}

{12,14,34} {12,13,23,34}

{12,23,24} {12,14,23,24}

{12,23,34} {12,14,24,34}

{12,24,34} {12,23,24,34}

{13,14,23} {13,14,23,34}

{13,14,24} {13,14,24,34}

{13,23,24} {13,23,24,34}

{13,23,34} {14,23,24,34}

{13,24,34}

{14,23,24}

{14,23,34}

{14,24,34}

MATHEMATICA

Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]], {n, 0, 5}]

CROSSREFS

Column k=1 of A327366.

The non-covering version is A245797.

The unlabeled version is A324693.

The generalization to set-systems is A327229.

BII-numbers of set-systems with minimum degree 1 are A327105.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 01 2019

STATUS

approved