Counting Set Covers and Split Graphs

A bijection between split graphs and minimal covers of a set by subsets is presented. As the enumeration problem for such minimal covers has been solved, this implies that split graphs can also be enumerated. 1 Motivation A split graph is a chordal graph with a chordal complement. It is straightforward to recognize split graphs, and therefore to compute the numbers of split graphs on a small number of vertices, as shown in Table 1. Whenever such a table is given, it is to be understood that they contain numbers of pairwise non-isomorphic objects, rather than ‘labeled’ objects. The numbers in Table 1 form sequence A48194 in [5], which is an online database of interesting sequences of integers (see also [4]). One of the aims of this database is to permit researchers who encounter a sequence to determine whether it has occurred before, and in what context, thereby exposing possibly unexplored connections. A k-cover of an n-set N is a collection of k subsets of N whose union is N . A k-cover is minimal if no sub-collection also covers N . Clarke [1] gives an expression for the number of minimal k-covers of an n-set (where again it is to be understood that the numbers refer to the number of pairwise non-isomorphic objects). Using this formula, Michael Somos (private communication) computed the total number of minimal covers of an n-set and using [5] observed that for n ≤ 11 (the limit of the sequence known at that time), this number was equal to the number of split graphs on n vertices. The current paper shows that this is no coincidence by proving the following result: