Obstructions to Faster Diameter Computation: Asteroidal Sets

Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than $\alpha$ pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every $m$-edge graph in $Ext_{\alpha}$ can be computed in deterministic ${\cal O}(\alpha^3 m^{3/2})$ time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive $+1$-approximation of all vertex eccentricities in deterministic ${\cal O}(\alpha^2 m)$ time. This is in sharp contrast with general $m$-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in ${\cal O}(m^{2-\epsilon})$ time for any $\epsilon > 0$. As important special cases of our main result, we derive an ${\cal O}(m^{3/2})$-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an ${\cal O}(k^3m^{3/2})$-time algorithm for this problem on graphs of asteroidal number at most $k$. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions

Coning-off CAT(0) cube complexes

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical hyperbolicity of $C'(1/4)-T(4)$ small cancellation quotients of free products.Comment: 45 pages, 13 figures. Comments are welcom

Group actions on injective spaces and Helly graphs

These are lecture notes for a minicourse on group actions on injective spaces and Helly graphs, given at the CRM Montreal in June 2023. We review the basics of injective metric spaces and Helly graphs, emphasizing the parallel between the two theories. We also describe various elementary properties of groups actions on such spaces. We present several constructions of injective metric spaces and Helly graphs with interesting actions of many groups of geometric nature. We also list a few exercises and open questions at the end.Comment: Comments are welcome! v2: some references adde

Helly groups

Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graphical C(4)$-$T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type $C_n$ are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right-angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier

Beyond Helly graphs: the diameter problem on absolute retracts

Characterizing the graph classes such that, on $n$-vertex $m$-edge graphs in the class, we can compute the diameter faster than in ${\cal O}(nm)$ time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph $H$ of a graph $G$ is called a retract of $G$ if it is the image of some idempotent endomorphism of $G$. Two necessary conditions for $H$ being a retract of $G$ is to have $H$ is an isometric and isochromatic subgraph of $G$. We say that $H$ is an absolute retract of some graph class ${\cal C}$ if it is a retract of any $G \in {\cal C}$ of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized $\tilde{\cal O}(m\sqrt{n})$ time. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of $k$-chromatic graphs, for every fixed $k \geq 3$. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

The simplicial boundary of a CAT(0) cube complex

For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_\Delta\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare $\partial_\Delta\mathbf X$ to the Roller, visual, and Tits boundaries of $\mathbf X$ and give conditions under which the natural CAT(1) metric on $\partial_\Delta\mathbf X$ makes it (quasi)isometric to the Tits boundary. $\partial_\Delta\mathbf X$ allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph} $\Gamma\mathbf X$, which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in $\mathbf X$ using $\partial_\Delta\mathbf X$. Finally, we rephrase the rank-rigidity theorem of Caprace-Sageev in terms of group actions on $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$ and state characterizations of cubulated groups with linear divergence in terms of $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$.Comment: Lemma 3.18 was not stated correctly. This is fixed, and a minor adjustment to the beginning of the proof of Theorem 3.19 has been made as a result. Statements other than 3.18 do not need to change. I thank Abdul Zalloum for the correction. See also: arXiv:2004.01182 (this version differs from previous only by addition of the preceding link, at administrators' request