Generalised Dualities and Finite Maximal Antichains

We fully characterise the situations where the existence of a homomorphism from a digraph G to at least one of a finite set of directed graphs is determined by a finite number of forbidden subgraphs. We prove that these situations, called generalised dualities, are characterised by the non-existence of a homomorphism to G from a finite set of forests. Furthermore, we characterise all finite maximal antichains in the partial order of directed graphs ordered by the existence of homomorphism. We show that these antichains correspond exactly to the generalised dualities. This solves a problem posed in [1]. Finally, we show that it is NP-hard to decide whether a finite set of digraphs forms a maximal antichain.

[1]  Saharon Shelah,et al.  On the order of countable graphs , 2003, Eur. J. Comb..

[2]  Emo Welzl.  Color-Families are Dense , 1982, Theor. Comput. Sci..

[3]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[4]  Albert Atserias,et al.  On digraph coloring problems and treewidth duality , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[5]  Peter Jeavons,et al.  Constraint Satisfaction Problems and Finite Algebras , 2000, ICALP.

[6]  Xuding Zhu,et al.  Duality and Polynomial Testing of Tree Homomorphisms , 1996 .

[7]  Jaroslav Nesetril,et al.  On classes of relations and graphs determined by subobjects and factorobjects , 1978, Discret. Math..

[8]  Jaroslav Nesetril,et al.  Duality Theorems for Finite Structures (Characterising Gaps and Good Characterisations) , 2000, J. Comb. Theory, Ser. B.

[9]  Jaroslav Nesetril,et al.  On maximal finite antichains in the homomorphism order of directed graphs , 2003, Discuss. Math. Graph Theory.

[10]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[11]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[12]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[13]  Peter Jeavons,et al.  The Complexity of Constraint Languages , 2006, Handbook of Constraint Programming.

[14]  Peter Jeavons,et al.  On the Algebraic Structure of Combinatorial Problems , 1998, Theor. Comput. Sci..

[15]  Claude Tardif,et al.  A Characterisation of First-Order Constraint Satisfaction Problems , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).