There are several examples of interpreting the Catalan numbers as non-nesting or non-crossing matchings of some graph. My question is: Is there a family of graphs $G_1,G_2,\dotsc$ with the number of vertices growing linearly, such that the number of matchings (not complete matchings) of $G_i$ is the $i$th Catalan number?

I have checked Richard Stanley's book on Catalan numbers, but such an interpretation is not there. The closest I found is a graph, where one counts the number of complete matchings, but this is not what I am after.