Quasi-median Graphs Research Articles

Translation lengths in crossing and contact graphs of quasi-median graphs

Given a quasi-median graph X , the crossing graph \Delta X and the contact graph \Gamma X are natural hyperbolic models of X . In this article, we show that the asymptotic translation length in \Delta X or \Gamma X of an isometry of X is always rational. Moreover, if X is hyperbolic, these rational numbers can be written with denominators bounded above uniformly; this is not true in full generality. Finally, we show that, if the quasi-median graph X is constructible in some sense, then there exists an algorithm computing the translation length of every computable isometry. Our results encompass contact graphs in CAT(0) cube complexes and extension graphs of right-angled Artin groups.

Special cube complexes revisited: a quasi-median generalization

AbstractIn this article, we generalize Haglund and Wise’s theory of special cube complexes to groups acting on quasi-median graphs. More precisely, we define special actions on quasi-median graphs, and we show that a group which acts specially on a quasi-median graph with finitely many orbits of vertices must embed as a virtual retract into a graph product of finite extensions of clique-stabilizers. In the second part of the article, we apply the theory to fundamental groups of some graphs of groups calledright-angled graphs of groups.

Groups acting on quasi-median graphs. An introduction

Quasi-median graphs have been introduced by Mulder in 1980 as a generalisation of median graphs, known in geometric group theory to naturally coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author showed that quasi-median graphs may be useful to study groups as well. In the present paper, we propose a gentle introduction to the theory of groups acting on quasi-median graphs.

Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products

In this paper we study group actions on quasi-median graphs, or "CAT(0) prism complexes", generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a quasi-median graph X and define the contact graph \mathcal{C}X for these hyperplanes. We show that \mathcal{C}X is always quasi-isometric to a tree, generalising a result of Hagen [18], and that under certain conditions a group action G \curvearrowright X induces an acylindrical action G \curvearrowright \mathcal{C}X , giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto [5]. As an application, we exhibit an acylindrical action of a graph product on a quasi-tree, generalising results of Kim and Koberda for right-angled Artin groups [20, 21]. We show that for many graph products G , the action we exhibit is the "largest" acylindrical action of G on a hyperbolic metric space. We use this to show that the graph products of equationally noetherian groups over finite graphs of girth \geq 6 are equationally noetherian, generalising a result of Sela [27].

Computing the blocks of a quasi-median graph

Quasi-median graphs are a tool commonly used by evolutionary biologists to visualise the evolution of molecular sequences. As with any graph, a quasi-median graph can contain cut vertices, that is, vertices whose removal disconnect the graph. These vertices induce a decomposition of the graph into blocks, that is, maximal subgraphs which do not contain any cut vertices. Here we show that the special structure of quasi-median graphs can be used to compute their blocks without having to compute the whole graph. In particular we present an algorithm that, for a collection of n aligned sequences of length m, can compute the blocks of the associated quasi-median graph together with the information required to correctly connect these blocks together in run time O(n2m2), independent of the size of the sequence alphabet. Our primary motivation for presenting this algorithm is the fact that the quasi-median graph associated to a sequence alignment must contain all most parsimonious trees for the alignment, and therefore precomputing the blocks of the graph has the potential to help speed up any method for computing such trees.

Replacing cliques by stars in quasi-median graphs

For a multi-set Σ of splits (bipartitions) of a finite set X, we introduce the multi-split graph G( Σ). This graph is a natural extension of the Buneman graph. Indeed, it is shown that several results pertaining to the Buneman graph extend to the multi-split graph. In addition, in case Σ is derived from a set R of partitions of X by taking parts together with their complements, we show that the extremal instances where R is either strongly compatible or strongly incompatible are equivalent to G( Σ) being either a tree or a Cartesian product of star trees, respectively.

Replacing cliques by stars in quasi-median graphs*1

For a multi-set Σ of splits (bipartitions) of a finite set X , we introduce the multi-split graph G ( Σ ). This graph is a natural extension of the Buneman graph. Indeed, it is shown that several results pertaining to the Buneman graph extend to the multi-split graph. In addition, in case Σ is derived from a set R of partitions of X by taking parts together with their complements, we show that the extremal instances where R is either strongly compatible or strongly incompatible are equivalent to G ( Σ ) being either a tree or a Cartesian product of star trees, respectively.

Quasi-median graphs, their generalizations, and tree-like equalities

Three characterizations of quasi-median graphs are proved, for instance, they are partial Hamming graphs without convex house and convex Q 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. Quasi-semimedian graphs are characterized which yields an additional characterization of quasi-median graphs. Two equalities for quasi-median graphs are proved. One of them asserts that if α i , i≥0, denotes the number of induced Hamming subgraphs of a quasi-median graph, then ∑ i≥0 (−1) i α i=1 . Finally, an Euler-type formula is derived for graphs that can be obtained by a sequence of connected expansions from K 1.

Fiber-complemented graphs II. Retractions and endomorphisms

A fiber-complemented graph is a graph for which the inverse image of every prefiber (or gated set) by any projection map onto a prefiber is a prefiber. In this paper, we continue the study of these graphs and establish a retraction theorem and fixed point properties for endomorphisms. Adding the notion of mooring (these are particular retractions of a graph onto its prefibers) to the tools introduced in Part I of this work (Discrete Math. 226 (2001) 107), we show that a fiber-complemented graph whose elementary prefibers induce moorable graphs is a retract of a Cartesian product of elementary moorable graphs. Then we deduce that under some conditions of compacticity, the elements of every commuting family of endomorphisms of a moorable pre-median graph strictly stabilize a nonempty finite pre-median subgraph ( pre-median graphs are particular instances of weakly modular graphs which are fiber-complemented). These results give generalizations of analogous properties related to median graphs, quasi-median graphs, pseudo-median graphs and weakly median graphs.

On the Natural Imprint Function of a Graph

In this paper, some characterizations of median and quasi-median graphs are extended to general isometric subgraphs of Cartesian products using the concept of an imprint function as introduced by Tardif. This extends the well known concepts of medians in median graphs as well as imprints in quasi-median graphs. We introduce absolute C-median graphs in analogy to absolute retracts, and derive a connection with the canonical isometric embedding of graphs into Cartesian products. Absolute C-median graphs strictly include classes of irreducible graphs and absolute (weak) retracts as well as many median-like classes, such as weakly median graphs, pre-median graphs, and weakly modular graphs. New characterizations of quasi-median graphs and of median graphs are obtained along the way. Finally, we propose a conjecture on the amalgamation procedure for absolute C-median graphs, and prove the fixed box theorem for this class modulo the conjecture.

Quasi-median graphs from sets of partitions

In studies of molecular evolution, one is typically confronted with the task of inferring a phylogenetic tree from a set X of sequences of length n over a finite alphabet Λ. For studies that invoke parsimony, it has been found helpful to consider the quasi-median graph generated by X in the Hamming graph Λ n . Although a great deal is already known about quasi-median graphs (and their algebraic counterparts), little is known about the quasi-median generation in Λ n starting from a set X of vertices. We describe the vertices of the quasi-median graph generated by X in terms of the coordinatewise partitions of X. In particular, we clarify when the generated quasi-median graph is the so-called relation graph associated with X. This immediately characterizes the instances where either a block graph or the total Hamming graph is generated.

Partial Hamming graphs and expansion procedures

Structural properties of isometric subgraphs of Hamming graphs are presented, generalizing certain results on quasi-median graphs. Consequently, a relation on the edge set of a graph which is closely related to Winkler–Djoković's relation Θ is introduced and used for a characterization of isometric subgraphs of Hamming graphs. Moreover, some results considering semi-median graphs and expansions on isometric subgraphs of hypercubes are extended to general non-bipartite case.

Fiber-complemented graphs — I: structure and invariant subgraphs

A subset A of a metric space ( X, d) is a prefiber (or a gated set) of X if, for every x∈ X, there exists y∈ A such that d( x, z)= d( x, y)+ d( y, z) for every z∈ A. In a graph endowed with the structure of metric space associated with the geodesic distance, the prefibers induce a convexity. In this paper, we introduce the class of fiber-complemented graphs for which the inverse image of every prefiber, by any projection map, onto a prefiber is a prefiber. From this property we deduce: (1) a procedure of construction by amalgamation or by expansion where the minimal prefibers with respect to inclusion (called elementary prefibers) work like building stones and depend on each class of graphs; (2) a theorem of canonical isometric embedding into a Cartesian product of graphs whose factors, called elementary graphs, are induced by elementary prefibers. Then, we study the topology of these graphs, with respect to the convexity induced by the prefibers, in order to determine its subbase and to characterize compact fiber-complemented graphs. From this, we deduce that every fiber-complemented graph without isometric ray contains a Cartesian product of elementary graphs which is invariant under every automorphism. As a consequence, this theory gives a global approach to obtain several previous results related to median graphs, quasi-median graphs, pseudo-median graphs, weakly median graphs, bridged graphs, and extend them to the infinite case.

Decomposition andl1-Embedding of Weakly Median Graphs

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 not containing K4or K1,1,3as an induced subgraph. As a consequence one obtains that every finite weakly median graph is l1-embeddable, that is, it embeds as a metric subspace into some Rnequipped with the 1-norm.

On clique-gated graphs

Hagauer and Klavžar (1996) introduced clique-gated graphs as a generalization of quasi-median graphs. In their paper they proposed a problem whether a clique-gated graph can include any induced C5. This note gives a positive answer constructing such a graph. Moreover, a graph on 15 vertices which is a subgraph of every clique-gated including an induced C5 is found.

Retracts of Infinite Hamming Graphs

A Hamming graph is a Cartesian product of complete graphs. We show that (finite or infinite) quasi-median graphs, which are a generalization of median graphs, are exactly the retracts of Hamming graphs. This generalizes a result of Bandelt (1984,J. Graph Theory8, 501–510) about median graphs, and improves a result of Wilkeit (1992,Discrete Math.102, 191–218) and of Chung, Graham, and Saks (1989,Combinatorica9, 111–131) about finite quasi-median graphs.

Clique-gated graphs

Clique-gated graphs form an extension of quasi-median graphs. Two characterizations of these graphs are given and some other structural properties are obtained. An O( nm) algorithm is presented which recognizes clique-gated graphs. Here n and m denote the numbers of vertices and edges of a given graph, respectively.

Invariant Hamming graphs in infinite quasi-median graphs

It is shown that a quasi-median graph G without isometric infinite paths contains a Hamming graph (i.e., a cartesian product of complete graphs) which is invariant under any automorphism of G, and moreover if G has no infinite path, then any contraction of G into itself stabilizes a finite Hamming graph.

A Helly theorem in weakly modular space

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 the clique number of the associated graph. This result thus entails a Helly theorem for quasi-median graphs, pseudo-modular graphs, and bridged graphs.

Cartesian products of block graphs

Hamming graphs (being the Cartesian products of complete graphs) are known to be the quasi-median graphs not containing the 3-vertex path as a convex subgraph. Similarly, the Cartesian products of trees have been characterized among median graphs by a single forbidden convex subgraph. In this note a common generalization of these two results is given that characterizes the Cartesian products of block graphs.