On the spectrum of the closed-set lattice of a graph

Let C(G) be the closed-set lattice of a graph G, and let l(f) denote the length of a chain r in C( G). The spectrum of £( G) is defined as the set S(£(G)) = {/(r) I r is a maximal chain in £(G)}. For every nontrivial graph G,S(£(G)) is a finite set of natural numbers greater than one. We prove in this paper that (*) for any finite set A of natural numbers greater than one, there exists a graph G such that S(£( G» = A. A set 5 of vertices of a graph G is said to be k-independent if d( u., v) 2:: k for all distinct members u, v in S. A k-independent set of G is said to be maximal if it is not properly contained in any k-independent set of G. To prove the result (*), we first establish the following: Given any finite set B of natural numbers, there exists a graph G such that {I 5 II 5 is a maximal 3-independent set of G} = B.