Graphs with $$G^p$$-connected medians

Discrete Applied Mathematics, 2000

A median of a k-tuple = (x 1; : : : ; x k ) of vertices of a ÿnite connected graph G is a vertex x for which k i=1 d(x; xi) is minimum, where d is the geodesic metric on G. The function M with domain the set of all k-tuples with k ¿ 0 and deÿned by M ( ) = {x | x is a median of } is called the median function on G. In this paper a new characterization of the median function is given for G a median graph. This is used to give a characterization of the median function on median semilattices. ? 2000 Elsevier Science B.V. All rights reserved. MSC: primary 05C12; 05C75; secondary 06A12; 90A08

2007

A profile on a graph G is a finite sequence of vertices of G. The remoteness of a vertex u is the sum of distances to the vertices of the profile. The set of vertices that maximize (minimize) the remoteness is the antimedian (median) set of the profile. It is proved that for every connected graph G there exists a graph H such that G is a convex subgraph of H and V (G) is the antimedian set of the profile consisting of V (H). Using linear programming it is also proved that for an arbitrary graph G and S ⊆ V (G) it can be decided in polynomial time whether S is the antimedian set of some profile. Both results are extended to median sets as well. Graphs in which every antimedian set is connected are also considered.

SIAM Journal on Discrete Mathematics, 1999

Let M (m, n) be the complexity of checking whether a graph G with m edges and n vertices is a median graph. We show that the complexity of checking whether G is triangle-free is at most M (m, m). Conversely, we prove that the complexity of checking whether a given graph is a median graph is at most O(m log n) + T (m log n, n), where T (m, n) is the complexity of finding all triangles of the graph. We also demonstrate that, intuitively speaking, there are as many median graphs as there are triangle-free graphs. Finally, these results enable us to prove that the complexity of recognizing planar median graphs is linear.

Discussiones Mathematicae Graph Theory, 2010

The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join

Discrete Mathematics, 1993

A notion of parallelism is defined in finite median graphs and a number of properties about geodesics and the existence of cubes are obtained. Introducing sites as a double structure of partial order and graph on a set, it is shown that all median graphs can be constructed from sites and, in fact, that the categories of sites and pointed median graphs are equivalent, generalizing Birkhoff's duality.

Algorithmica, 2010

The median (antimedian) set of a profile π = (u 1 , . . . , u k ) of vertices of a graph G is the set of vertices x that minimize (maximize) the remoteness i d(x, u i ). Two algorithms for median graphs G of complexity O(n idim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes.

Networks, 2010

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 H, respectively, and that dH (G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H.

Discrete Mathematics, 1998

Let G be a median graph on n vertices and m edges and let k be the number of equivalence classes of the Djoković's relation Θ defined on the edge-set of G. Then 2n − m − k ≤ 2. Moreover, 2n − m − k = 2 if and only if G is cube-free.

Discrete Mathematics, 2003

A graph is distance-hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph. In this paper, we study metric properties of distance-hereditary graphs. In particular, we determine the structures of centers and medians of distance-hereditary and related graphs. The relations between eccentricity, radius, and diameter of such graphs are also investigated.

Discussiones Mathematicae Graph Theory, 2005

Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G × P 3 is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G × K 2 are proved, and it is shown that the only nonbipartite quasimedian direct product is K 3 × K 3 .

Discrete Mathematics, 2011

We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends. For cubic median graphs G the condition of linear growth can be weakened to the condition that G has two ends. For higher degree the relaxation to twoended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.

Discrete Applied Mathematics, 2013

A profile π = (x 1 ,. .. , x k), of length k, in a finite connected graph G is a sequence of vertices of G, with repetitions allowed. A median x of π is a vertex for which the sum of the distances from x to the vertices in the profile is minimum. The median function finds the set of all medians of a profile. Medians are important in location theory and consensus theory. A median graph is a graph for which every profile of length 3 has a unique median. Median graphs have been well studied, possess a beautiful structure and arise in many arenas, including ternary algebras, ordered sets and discrete distributed lattices. They have found many applications, for instance in location theory, consensus theory and mathematical biology. Trees and hypercubes are key examples of median graphs. We establish a succinct axiomatic characterization of the median procedure on median graphs, settling a question posed implicitly by McMorris, Mulder and Roberts in 1998 [19]. We show that the median procedure can be characterized on the class of all median graphs with only three simple and intuitively appealing axioms, namely anonymity, betweenness and consistency. Our axiomatization is tight in the sense that each of these three axioms is necessary. We also extend a key result of the same paper, characterizing the median function for profiles of even length on median graphs.

Surveys on Discrete and Computational Geometry, 2008

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 l 1-graphs. Several kinds of l 1-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 graph classes reported in this survey.

2011

A profile = (x1, ..., xk), of length k, in a finite connected graph G is a sequenceof vertices of G, with repetitions allowed. A median x of is a vertex for whichthe sum of the distances from x to the vertices in the profile is minimum. Themedian function finds the set of all medians of a profile. Medians are important inlocation theory and consensus theory. A median graph is a graph for which everyprofile of length 3 has a unique median. Median graphs are well studied. Theyarise in many arenas, and have many applications.We establish a succinct axiomatic characterization of the median procedure onmedian graphs. This is a simplification of the characterization given by McMorris,Mulder and Roberts [17] in 1998. We show that the median procedure can be characterizedon the class of all median graphs with only three simple and intuitivelyappealing axioms: anonymity, betweenness and consistency. We also extend a keyresult of the same paper, characterizing the median function for profiles...

Let G be a bipartite graph. In this paper we consider the two kind of location problems namely p-center and p-median problems on bipartite graphs. The p-center and p-median problems asks to find a subset of vertices of cardinality p, so that respectively the maximum and sum of the distances from this set to all other vertices in G is minimized. For each case we present some properties to find exact solutions.