Approximating number of colorings seems to be easy on minor-excluded graphs using algorithm by Jung/Shah. What are other examples of problems that are hard on general graphs but easy on minor-excluded graphs?
Update 10/24 It seems to follow Grohe's results that formula that is FPT to test on bounded-treewidth graphs is FPT to test on minor excluded graphs. Now the question is -- how does it relate to tractability of counting satisfying assignments of such formula?
The above statement is false. MSOL is FPT on bounded tree-width graphs, however 3-colorability is NP-complete on planar graphs which are minor-excluded.