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We provide a correspondence between the subjects of duality and density in classes of finite relational structures. The purpose of du-ality is to characterise the structures C that do not admit a homo-morphism into a given target B by the existence of a homomorphism from a structure A into C. Density is the order-theoretic property of containing no covers(More)
We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant , the greatest reduced average density (grad) of G with rank r, ∇ r (G). For these classes we prove the existence of several partition results such as the existence of low(More)
We define the notions tree-depth and upper chromatic number of a graph and show their relevance to local–global problems for graph partitions. In particular we show that the upper chromatic number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs. The rich interplay of these(More)
Bor uvka presented in 1926 the ÿrst solution of the Minimum Spanning Tree Problem (MST) which is generally regarded as a cornerstone of Combinatorial Optimization. In this paper we present the ÿrst English translation of both of his pioneering works. This is followed by the survey of development related to the MST problem and by remarks and historical(More)
Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). These classes are also characterized by the existence of several partition results such as the existence of low(More)
The oriented chromatic number o(H) of an oriented graph H is deened as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0. The oriented chromatic number o(G) of an undirected graph G is then deened as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) deened as(More)
<i>Classes of graphs with bounded expansion</i> have been introduced in [15], [12]. They generalize both proper minor closed classes and classes with bounded degree.For any class with bounded expansion C and any integer p there exists a constant N(C,p) so that the vertex set of any graph G &#8712; C may be partitioned into at most N(C,p) parts, any i(More)