T-Theory is the name that we adopt for the theory of trees, injective envelopes of metric spaces, and all of the areas that are connected with these topics, which has been developed over the last 10-15 years in Bielefeld. Its motivation was originally -- and still is to a large extent -- the development of mathematical tools for reconstructing phylogenetic trees. T-theory expanded considerably when its relationships with the theory of affine buildings, valuated matroids, and decompositions of metrics were discovered. In this paper, we give a brief introduction to this theory, which we hope will serve as a useful reference to some of the main results, and also as a guide for further investigations into what T-theory has to offer.

In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G�K � is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT(0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1- or 2-fellow traveler property. © 2000 Academic Press 1.

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enriched-categorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the ffl-ball topology; 3. lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized ultrametric spaces. Restricted to the special cases of preorders and ordinary ultrametric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the ffl-ball topology; 3. lower, upper, and convex powerdomains, and the powerdomain of compact subsets. Interestingly, all constructions are formulated in terms of (an ultrametric version of) the Yoneda (1954) lemma.

The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l1-graphs. Several kinds of l1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆-)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or tree-like graphs such as distance-hereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the

The paper describes combinatorial modeling of decomposable systems: hierarchical system models, system components, design alternatives (DAs) for system components and their interconnection (Is), estimates of DAs and Is, changes of the systems. We point out some basic combinatorial operations: combinatorial description and presentation; analysis and evaluation; revealing of bottlenecks by elements; comparison of system versions; synthesis of composite DAs; modification (e.g., improvement, adaptation). The investigation is based on hierarchical morphological multicriteria design (HMMD) which involves an examination of the following: the design of hierarchical system model, the generation and assessment DAs and Is, composing of composite DAs, and improvements on the base of bottlenecks. Our list of support combinatorial problems is the following: design of hierarchical system models; multicriteria ranking (ordinal assessment), morphological clique (synthesis), etc....

We give simple algorithmic proofs of some theorems of Papernov (1976) and Karzanov (1985,1990) on the packing of metrics by cuts. 1.

. This paper is concerned with a new distance in undirected graphs with weighted edges, which gives new insights into the structure of all minimum spanning trees of a graph. This distance is a generalized one, in the sense that it takes values in a certain Heyting semigroup. More precisely, it associates with each pair of distinct vertices in a connected component of a graph the set of all paths joining them in the minimum spanning trees of that component. A partial order and an addition of these sets of paths are defined. We show how general algorithms for path algebra problems can be used to compute the generalized distance. Some theoretical problems concerning this distance are formulated. The main application of our generalized distance is related to recent clustering procedures. Given a connected graph with weighted edges and certain vertices labeled as centers, we define a centered forest to be a spanning forest with exactly one center in each tree component. A partition of the v...

Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity- the k-colouring problem is polynomial time solvable when k ≤ 2, and NP-complete when k ≥ 3. Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction

Let H be a graph and k ≥ 3. A near-unanimity function of arity k is a mapping g from the k-tuples over V (H) toV (H) such that g(x1,x2,...,xk) is adjacent to g(x ′ 1,x ′ 2,...,x ′ k) whenever xix ′ i ∈ E(H) for each i =1, 2,...,k, and g(x1,x2,...,xk) =a whenever at least k − 1of the xi’s equal a. Feder and Vardi proved that, if a graph H admits a near-unanimity function, then the homomorphism extension (or retraction) problem for H is polynomial time solvable. We focus on near-unanimity functions on reflexive graphs. The best understood are reflexive chordal graphs H: they always admit a near-unanimity function. We bound the arity of these functions in several ways related to the size of the largest clique and the leafage of H, and we show that these bounds are tight. In particular, it will follow that the arity is bounded by n − √ n + 1, where n = |V (H)|. We investigate substructures forbidden for reflexive graphs that admit a near-unanimity function. It will follow, for instance, that no reflexive cycle of length at least four admits a near-unanimity function of any arity. However, we exhibit nonchordal graphs which do admit near-unanimity functions. Finally, we characterize graphs which admit a conservative near-unanimity function. This characterization has been predicted by the results of Feder, Hell, and Huang. Specifically, those results imply that, if P ̸ = NP, the graphs with conservative near-unanimity functions are precisely the so-called bi-arc graphs. We give a proof of this statement without assuming P ̸ = NP.