Before revision:  A seminal result of Grohe, Schwentick, and Segoufin (STOC 2004 ) demonstrates that when restricted to graphs, a class of conjunctive queries can be evaluated in polynomial time if and only if it has bounded treewidth.
After revision:  A seminal result of Grohe, Schwentick, and Segoufin (STOC 2004 ) demonstrates that for every class G of graphs, the evaluation of all conjunctive queries whose underlying graph is in G is tractable if, and only if it has bounded treewidth.