Thanks Daniel, your suggestion worked flawlessly. It compelled me to upgrade from version 1.10 (which did not appear to offer the #tag: parser function), to 1.12.

Since my template only uses the global parameter SUBPAGENAMEE, in an #ifeq: function, my template call is as simple as

{{#tag: graphviz | {{Template:View_structure}} }}

That little gem will save dozens of copies of nearly identical graphviz code...