Connected graphs containing a given connected graph as a unique greatest common subgraph

Those connected graphsG are determined for which there exist nonisomorphic connected graphs of equal size containingG as a unique greatest common subgraph. Analogous results are also obtained for weakly connected and strongly connected digraphs, as well as for induced subgraphs and induced subdigraphs.This research was supported by a Western Michigan University faculty research fellowship.This research was supported in part by a Western Michigan University research assistantship from the Graduate College and the College of Arts and Sciences.