Let K be a class of finite graphs and F = {F1, F2, ..., Fm} be a set of finite graphs. Then, K is said to have finite-duality if there exists a graph U in K such that for any graph G in K, G is homomorphic to U if and only if Fi is not homomorphic to G, for all i = 1, 2, ..., m. Nešetřil asked in [J. Nešetřil, Homonolo Combinatorics Workshop, Nova Louka, Czech Rep., (2003)] if non-trivial examples can be found. In this note, we answer this positively by showing classes containing arbitrary long anti-chains and yet having the finite-duality property.

Let K be a class of graphs. A pair (F,U) is a finite duality in K if U∈K, F is a finite set of graphs, and for any graph G in K we have G≤U if and only if F≤≰G for all F∈F, where "≤" is the homomorphism order. We also say U is a dual graph in K. We prove that the class of planar graphs has no finite dualities except for two trivial cases. We also prove that the class of toroidal graphs has no 5-colorable dual graphs except for two trivial cases. In a sharp contrast, for a higher genus orientable surface S we show that Thomassen's result (Thomassen, 1997 [17]) implies that the class, G(S), of all graphs embeddable in S has a number of finite dualities. Equivalently, our first result shows that for every planar core graph H except K1 and K4, there are infinitely many minimal planar obstructions for H-coloring (Hell and Nešetřil, 1990 [4]), whereas our later result gives a converse of Thomassen's theorem (Thomassen, 1997 [17]) for 5-colorable graphs on the torus.