Minimal universal and dense minor closed classes

We study the homomorphism (coloring) order induced on minor closed classes. In [J. Hubička, J. Nešetřil, Finite Paths are Universal, ITI Series 2003-129, Charles University, 2003. Order (in press)], the minor closed class P of directed paths is shown to be universal and in [J. Nešetřil, X. Zhu, Path homomorphisms, Proc. Comb. Phil. Soc. (1996) 207–220], P is shown to contain a dense subset. In this note we prove that P is a unique minimal class of oriented graphs which is both universal and dense. Moreover, we show a dichotomy result for any minor closed class K of directed trees: K is either universal or it is well-quasi-ordered (wqo). We also prove structure theorems about series–parallel graphs (SPG), in an attempt to determine the minimal universal and dense minor closed classes of undirected graphs. We show the non-existence of universal classes in certain subclasses of SPG. Also for basic graphs in the class of SPG, we show that there is a linear time algorithm that decides whether such a graph is core or not. We also give a constructive description of arbitrary 2-connected graphs in SPG.