The relationship between two important problems in pattern recognition using attributed relational graphs, the maximum common subgraph and the minimum common supergraph of two graphs, is established by means of simple constructions, which allow to obtain the maximum common subgraph from the minimum common supergraph, and vice versa. On this basis, a new graph distance metric is proposed for measuring similarities between objects represented by attributed relational graphs. The proposed metric can be computed by a straightforward extension of any algorithm that implements error-correcting graph matching, when run under an appropriate cost function, and the extension only takes time linear in the size of the graphs.