On a construction of critical graphs which are not sensitive

Abstract

Let ℒ(G) be theclosed-set lattice of a graphG. G issensitive if the following implication is always true for any graphG′: ℒ(G)≅ℒ(G′)⇒(G)≅G′G iscritical if ℒ(G)≇ℒ(G-e) for anye inE(G) and ℒ(G)≇ℒ(G+e) for anye in$\left( {\bar G} \right)$ where$\bar G$ is the complement ofG. Every sensitive graph is, a fortiori, critical. Is every critical graph sensitive? A negative answer to this question is given in this note.