  In the field of parameterized complexity theory, the study of graph width
measures has been intimately connected with the development of width-based
model checking algorithms for combinatorial properties on graphs. In this work,
we introduce a general framework to convert a large class of width-based
model-checking algorithms into algorithms that can be used to test the validity
of graph-theoretic conjectures on classes of graphs of bounded width. Our
framework is modular and can be applied with respect to several well-studied
width measures for graphs, including treewidth and cliquewidth.
  As a quantitative application of our framework, we show that for several
long-standing graph-theoretic conjectures, there exists an algorithm that takes
a number $k$ as input and correctly determines in time double-exponential in
$k^{O(1)}$ whether the conjecture is valid on all graphs of treewidth at most
$k$. This improves significantly on upper bounds obtained using previously
available techniques.
