Obstructions to a small hyperbolicity in Helly graphs

Abstract

The δ-hyperbolicity of a graph is defined by a simple 4-point condition: for any four vertices u, v, w, and x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x), and d(u, x) + d(v, w) differ by at most 2δ ≥ 0. Hyperbolicity can be viewed as a measure of how close a graph is to a tree metrically; the smaller the hyperbolicity of a graph, the closer it is metrically to a tree. A graph G is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G there exists the smallest Helly graph H(G) into which G isometrically embeds (H(G) is called the injective hull of G) and the hyperbolicity of H(G) is equal to the hyperbolicity of G. Motivated by this, we investigate structural properties of Helly graphs that govern their hyperbolicity and identify three isometric subgraphs of the King-grid as structural obstructions to a small hyperbolicity in Helly graphs.

Injective hulls of various graph classes

“…Moreover, there are many graph parameters that are strongly related in Helly graphs, including so-called interval thinness, hyperbolicity, pseudoconvexity of disks, and size of the largest isometric subgraph in the form of a square rectilinear grid or a square king grid, among others (cf. [20,22]); in particular, a constant bound on any one of these parameters implies a constant bound on all others [20].…”

Fast deterministic algorithms for computing all eccentricities in (hyperbolic) Helly graphs

“…Graph hyperbolicity has attracted attention recently due to the empirical evidence that it takes small values in many real-world networks, such as biological networks, social networks, Internet application networks, and collaboration networks, to name a few (see, e.g., [2,3,9,58,62,67]). Furthermore, many special graph classes (e.g., interval graphs, chordal graphs, dually chordal graphs, AT-free graphs, weakly chordal graphs and many others) have constant hyperbolicity [2,4,14,21,34,38,59,70]. In fact, the dually chordal graphs and the C 4 -free Helly graphs are known to be proper subclasses of the 1-hyperbolic Helly graphs (this follows from results in [12,34]).…”

“…Furthermore, many special graph classes (e.g., interval graphs, chordal graphs, dually chordal graphs, AT-free graphs, weakly chordal graphs and many others) have constant hyperbolicity [2,4,14,21,34,38,59,70]. In fact, the dually chordal graphs and the C 4 -free Helly graphs are known to be proper subclasses of the 1-hyperbolic Helly graphs (this follows from results in [12,34]). Notice also that any graph is δ-hyperbolic for some δ ≤ diam(G)/2.…”

“…A minimal by inclusion Helly graph H which contains a given graph G as an isometric subgraph is unique and called the injective hull [57] or the tight span [42] of G. Polynomial-time recognition algorithms for the Helly graphs were presented in [6,28,61]. Several structural properties of these graphs were also identified in [6,7,8,18,28,29,30,33,34,64,65]. The dually chordal graphs are exactly the Helly graphs in which the intersection graph of balls is chordal, and they were studied independently from the general Helly graphs [11,12,41,31,33,68].…”

Beyond Helly graphs: the diameter problem on absolute retracts

“…Furthermore, a celebrated theorem in Metric Graph Theory is that every graph is an isometric (distance-preserving) subgraph of some Helly graph [44,64]. Other properties of Helly graphs were also thoroughly investigated in prior works [8,9,11,25,36,37,39,41,69,77,78]. In particular, as far as we are concerned here, there is a randomized Õ(m √ n)-time algorithm in order to compute the diameter within n-vertex m-edge Helly graphs with high probability [48].…”