22 research outputs found
Dualities and dual pairs in Heyting algebras
We extract the abstract core of finite homomorphism dualities using the
techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction
On infinite-finite duality pairs of directed graphs
The (A,D) duality pairs play crucial role in the theory of general relational
structures and in the Constraint Satisfaction Problem. The case where both
classes are finite is fully characterized. The case when both side are infinite
seems to be very complex. It is also known that no finite-infinite duality pair
is possible if we make the additional restriction that both classes are
antichains. In this paper (which is the first one of a series) we start the
detailed study of the infinite-finite case.
Here we concentrate on directed graphs. We prove some elementary properties
of the infinite-finite duality pairs, including lower and upper bounds on the
size of D, and show that the elements of A must be equivalent to forests if A
is an antichain. Then we construct instructive examples, where the elements of
A are paths or trees. Note that the existence of infinite-finite antichain
dualities was not previously known
Ramsey Properties of Permutations
The age of each countable homogeneous permutation forms a Ramsey class. Thus,
there are five countably infinite Ramsey classes of permutations
Splitting finite antichains in the homomorphism order
A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order
Adjoint functors in graph theory
We survey some uses of adjoint functors in graph theory pertaining to colourings, complexity reductions, multiplicativity, circular colourings and tree duality. The exposition of these applications through adjoint functors unifies the presentation to some extent, and also raises interesting questions
Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories
We survey results on Hedetniemi’s conjecture which are connected to adjoint functors in the “thin” category of graphs, and expose the obstacles to extending these results
Infinitely many minimal classes of graphs of unbounded clique-width
© 2017 Elsevier B.V. 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
Pivoting in Linear Complementarity: Two Polynomial-Time Cases
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