A characterization of halved cubes

European Journal of Combinatorics, 2009

It is proven that given G a subdivision of a clique K n (n ≥ 1), G is isometrically embeddable in a Hamming graph if and only if G is a partial cube or G = K n. The characterization for subdivided wheels is also obtained.

Analysis of Complex Networks, 2009

We survey various results on counting hypercubes and related problems. Since median graphs are built in a very special way from hypercubes, the number of hypercubes of different dimensions can also be considered as a measure of complexity for this class of graphs. Applications to phylogenetics are also mentioned.

European Journal of Combinatorics, 2003

Three characterizations of quasi-median graphs are proved, for instance, they are partial Hamming graphs without convex house and convex Q3− such that certain relations on their edge sets coincide. Expansion procedures, weakly 2-convexity, and several relations on ...

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.

arXiv (Cornell University), 2022

The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatization of the main classes of graphs occurring in Metric Graph Theory, in analogy to Tarski's axiomatization of Euclidean geometry. We provide such an axiomatization for weakly modular graphs and their principal subclasses (median and modular graphs, bridged graphs, Helly graphs, dual polar graphs, etc), basis graphs of matroids and even ∆-matroids, partial cubes and their subclasses (ample partial cubes, tope graphs of oriented matroids and complexes of oriented matroids, bipartite Pasch and Peano graphs, cellular and hypercellular partial cubes, almost-median graphs, netlike partial cubes), and Gromov hyperbolic graphs. On the other hand, we show that some classes of graphs (including chordal, planar, Eulerian, and dismantlable graphs), closely related with Metric Graph Theory, but defined in a combinatorial or topological way, do not allow such an axiomatization.

2011

Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost selfcentered graphs among median graphs and among chordal graphs. In the first case P 4 and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is much richer, despite the fact that their diameter is at most 3. We also discuss almost self-centered graphs among partial cubes and among k-chordal graphs, classes of graphs that generalize median and chordal graphs, respectively. Characterizations of almost self-centered graphs among these two classes seem elusive.

Discrete Applied Mathematics, 2009

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x ∈ V (G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(m log n) time whether G is a median graph with geodetic number 2.

ArXiv, 2017

$k$-point crossover operators and their recombination sets are studied from different perspectives. We show that transit functions of $k$-point crossover generate, for all $k>1$, the same convexity as the interval function of the underlying graph. This settles in the negative an open problem by Mulder about whether the geodesic convexity of a connected graph $G$ is uniquely determined by its interval function $I$. The conjecture of Gitchoff and Wagner that for each transit set $R_k(x,y)$ distinct from a hypercube there is a unique pair of parents from which it is generated is settled affirmatively. Along the way we characterize transit functions whose underlying graphs are Hamming graphs, and those with underlying partial cube graphs. For general values of $k$ it is shown that the transit sets of $k$-point crossover operators are the subsets with maximal Vapnik-Chervonenkis dimension. Moreover, the transit sets of $k$-point crossover on binary strings form topes of uniform oriented matroid of VC-dimension $k+1$. The Topological Representation Theorem for oriented matroids therefore implies that $k$-point crossover operators can be represented by pseudosphere arrangements. This provides the tools necessary to study the special case $k=2$ in detail.

A periphery transversal of a median graph G is introduced as a set of vertices that meets all the peripheral subgraphs of G. Using this concept, median graphs with geodetic number 2 are characterized in two ways. They are precisely the median graphs that contain a periphery transversal of order 2 as well as the median graphs for which there exists a profile such that the remoteness function is constant on G. Moreover, an algorithm is presented that decides in O(m log n) time whether a given graph G with n vertices and m edges is a median graph with geodetic number 2. Several additional structural properties of the remoteness function on hypercubes and median graphs are obtained and some problems listed.

For a partial cube (that is, an isometric subgraph of a hypercube) G, quotient graphs G # , G τ , and G ∼ have the equivalence classes of the Djoković-Winkler relation as the vertex set, while edges are defined in three different natural ways. Several results on these quotients are proved and the concepts are compared. For instance, for every graph G there exists a median graph M such that G = M τ . Triangle-free and complete quotient graphs are treated and it is proved that for a median graph G its τ -graph is triangle-free if and only if G contains no convex K 1,3 . Connectedness and the question of when quotients yield the same graphs are also treated.