On the expression of monadic second-order graph properties without quantifications over sets of edges

For graphs of degree at most some fixed integer, the same properties can be expressed by monadic second-order formulas with and without quantifications over sets of edges, with and without auxiliary orientations. Similar results hold for partial k-trees for fixed k, and for graphs of tree-width at most k. These results are related to the possibility of testing graph properties in polynomial time for graphs generated by context-free graph-grammars of various types