Constructions of sensitive graphs which are not strongly sensitive

Let G be a simple graph with vertex set V(G) and edge set E(G). For each a in V(G), let N(a) = {x E V(G) : ax E E(G)} be the set of neighbours of a. A subset S of V(G) is called a closed set of G if, for each pair of distinct elements u, b in S, N(a) ON(b) s S. Let Z’(G) be the family of closed sets of G, inclusive of the empty set 0. Evidently, Z(G) is closed under arbitrary intersection and it thus forms a lattice under set-inclusion (see Fig. 1). The lattice 2?(G), which was first introduced by Sauer (see [6] and also [3, 4, 5]), is called the closed-set lattice of the graph G. We shall now introduce, in terms of their closed-set lattices, various classes of graphs. A graph G is said to be minimally critical if 2(G) + _Y(G e) for each e in E(G), and maximally critical if Z(G) $ Z’(G + e) for any e in E(G), where G is the complement of G. We say that G is critical if G is both maximally and minimally critical. A graph G is said to be sensitive if for any graph G’ that 2?(G) = .Z(G’) implies G = G’. Suppose that G and G’ are graphs such that 2?(G) = Z(G’) un d er a lattice isomorphism @. It is easily seen that @ induces naturally a bijection 4 : V(G)+ V(G’) such that for each x in V(G), G(x) =x’ in V(G’) if and only if @({x}) = {x’} in .2(G)). W e call 4 the bijection induced by @. A graph G is said to be strongly sensitive if for any graph G’ and for any lattice isomorphism @: Z’(G) = Z(G’), the bijection Cp induced by Qi is a graph isomorphism of G onto G’. For a graph G, the following implications follow immediately from definitions: