A set of (labeled) graphs can be defined by a regular tree language and one regular string language for each possible edge label, as follows. For each tree t from the regular tree language the graph gr(t) has the same nodes as t (with the same labels), and there is an edge with label gamma from node x to node y if the string of labels of the nodes on the shortest path from x to y in t belongs to the regular string language for gamma. Slightly generalizing this definition scheme, we allow gr(t) to have only those nodes of t that have certain labels, and we allow a relabeling of these nodes. It is shown that in this way exactly the class of C-edNCE graph languages (generatedby C-edNCE graph grammars) is obtained, one of the largest known classes of context-free graph languages.