Products of graphs with their closed-set lattices

Abstract

Let L ( G), V( G) and Ḡ be, respectively, the closed-set lattice, vertex set, edge set and complement of a graph G. Any lattice isomorphism Φ: L(G)⋍ L(G′) induces a bijection Φ: V( G)→ V( G′) such that for each x in V( G), Φ( x)= x′ iff Φ({ x})={ x′}. A graph G is strongly sensitive if for any graph G′ and any lattice isomorphism Φ: L(G)⋍ L(G′) , the bijection Φ induced by Φ is a graph isomorphism of G onto G′. G is minimally critical if L(G) ∥ L(G-e) for each e in E( G), and maximally critical if L(G) ∥ L(G+e) for any e in E( G ̆ ) . In this paper, we prove that for any two nontrivial graphs G 1 and G 2, (1) G 1 × G 2 is maximally critical, and (2) G 1 × G 2 is strongly sensitive iff G 1 × G 2 is minimally critical. Necessary and sufficient conditions on G 1 such that G 1 × G 2 is strongly sensitive are also obtained.