Metric indexing is used to organize large databases of Attributed Graphs and define fast queries on these sets. We present a proposal where indexing is based on an m-tree in which routing nodes of the m-tree act as prototypes of the sub-cluster of Attributed Graphs that routing nodes represent. In the classical schemes, the information kept in the tree nodes of the subcluster is a selected Attr...

Three characterizations of quasi-median graphs are proved, for instance, they are partial Hamming graphs without convex house and convexQ 3 such that certain relations on their edge sets coincide. Expansion procedures, weakly 2-convexity, and several relations on the edge set of a graph are essential for these results. Quasisemimedian graphs are characterized which yields an additional characte...

Motivated by a dynamic location problem for graphs, Chung, Graham and Saks introduced a graph parameter called windex. Graphs of windex 2 turned out to be, in graph-theoretic language, retracts of hypercubes. These graphs are also known as median graphs and can be characterized as partial binary Hamming graphs satisfying a convexity condition. In this paper an O(n 3 2 logn) algorithm is present...

In this paper it is shown that a class of almost-median graphs that includes all planar almost-median graphs can be recognized in O(m log n) time, where n denotes the number of vertices and m the number of edges. Moreover, planar almost-median graphs can be recognized in linear time. As a key auxiliary result we prove that all bipartite outerplanar graphs are isometric subgraphs of the hypercub...

In this paper, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (also called the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels or, equivalently, that there exists a function f : N → N such that an event structure with degree ≤ n admits a labeling with at most f(n) labels. Our...

ÐIn object prototype learning and similar tasks, median computation is an important technique for capturing the essential information of a given set of patterns. In this paper, we extend the median concept to the domain of graphs. In terms of graph distance, we introduce the novel concepts of set median and generalized median of a set of graphs. We study properties of both types of median graph...

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r ≥ 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of ...

Weakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasi-median graphs as well as pseudo-median graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5-wheels, induced subgraphs of hyperoctahedra (alias cocktail party graphs), and 2-connected bridged graphs n...

Dendrons and their products admit a natural, continuous median operator. We prove that there exists a two-dimensional metric continuum with a continuous median operator, for which there is no median-preserving embedding in a product of finitely many dendrons. Our method involves ideas and results concerning graph colouring and abstract convexity. The main result answers a question in [16] negat...

The d-convex sets in a metric space are those subsets which include the metric interval between any two of its elements. Weak modularity is a certain interval property for triples of points. The d-convexity of a discrete weakly modular space X coincides with the geodesic convexity of the graph formed by the two-point intervals in X. The Helly number of such a space X turns out to be the same as...