The motivation for this paper is threefold. First, we study the connectivity properties of the homomorphism order of directed graphs, and more generally for relational structures. As opposed to the homomorphism order of undirected graphs (which has no non-trivial finite maximal antichains), the order of directed graphs has finite maximal * This research was partially supported by the EU Research Training Network COMB-STRU. † The third author's research is supported by grants from NSERC and ARP. ‡ The Institute for Theoretical Computer Science is supported as project 1M0021620808 by the Ministry of Education of the Czech Republic. antichains of any size. In this paper, we characterise explicitely all maximal antichains in the homomorphism order of directed graphs. Quite surprisingly, these maximal antichains correspond to generalised dualities. The notion of generalised duality is defined here in the full generality as an extension of the notion of finitary duality, investigated in [17]. Building upon the results of the cited paper, we fully characterise the generalised dualities. It appears that these dualities are determined by forbidding homomorphisms from a finite set of forests (rather than trees). Finally, in the spirit of [1], [12], [4] we shall characterise "generalised" Constraint Satisfaction Problems (defined also here) problems that are first order definable. These are again just generalised dualities corresponding to finite maximal antichains in the homomorphism order.

The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width.the clique-width of G. On the other hand, the clique-width of G is never smaller than the clique-width of any of its induced subgraphs [5]. This allows us to be restricted to hereditary classes, that is, those that are closed under taking induced subgraphs.One of the most remarkable outcomes of the graph minor project of Robertson and Seymour is the proof of Wagner's conjecture stating that the minor relation is a well-quasiorder [13]. This implies, in particular, that in the world of minor-closed graph classes there exist minimal classes of unbounded tree-width and the number of such classes is finite. In fact, there is just one such class (the planar graphs), which was shown even before the proof of Wagner's conjecture [12].In the world of hereditary classes the situation is more complicated, because the induced subgraph relation is not a well-quasi-order. It contains infinite antichains, and hence, there may exist infinite strictly decreasing sequences of graph classes with no minimal one. In other words, even the existence of minimal hereditary classes of unbounded clique-width is not an obvious fact. This fact was recently confirmed in [8]. However, whether the number of such classes is finite or infinite remained an open question. In the present paper, we settle this question by showing that the family of minimal hereditary classes of unbounded clique-width is infinite. Moreover, we prove that the same is true with respect to linear clique-width.The organisation of the paper is as follows. In the next section, we introduce basic notation and terminology. In Section 3, we describe a family of graph classes of unbounded clique-width and prove that infinitely many of them are minimal with respect to this property. In Section 4, we identify more classes of unbounded clique-width. Finally, Section 5 concludes the paper with a number of open problems.

We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty's least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris's highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP.

The paper [J. Balogh, B. Bollobás, D. Weinreich, A jump to the Bell number for hereditary graph properties, J. Combin. Theory Ser. B 95 (2005) 29-48] identifies a jump in the speed of hereditary graph properties to the Bell number B n and provides a partial characterisation of the family of minimal classes whose speed is at least B n . In the present paper, we give a complete characterisation of this family. Since this family is infinite, the decidability of the problem of determining if the speed of a hereditary property is above or below the Bell number is questionable. We answer this question positively by showing that there exists an algorithm which, given a finite set F of graphs, decides whether the speed of the class of graphs containing no induced subgraphs from the set F is above or below the Bell number. For properties defined by infinitely many minimal forbidden induced subgraphs, the speed is known to be above the Bell number.