Minimal universal and dense minor closed classes

  • Authors:

  • Jaroslav Nešetřil;Yared Nigussie

  • Affiliations:

  • Department of Applied Mathematics, Institute for Theoretical Computer Science (ITI), Charles University, Praha, Czech Republic;Department of Applied Mathematics, Institute for Theoretical Computer Science (ITI), Charles University, Praha, Czech Republic

  • Venue:
  • European Journal of Combinatorics - Special issue on Eurocomb'03 - graphs and combinatorial structures
  • Year:
  • 2006

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Abstract

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.