Fast Deterministic Algorithms for Computing All Eccentricities in (Hyperbolic) Helly Graphs

Abstract

We extend known results on chordal graphs and distance-hereditary graphs to much larger graph classes by using only a common metric property of these graphs. Specifically, a graph is called αi-metric (i ∈ N ) if it satisfies the following αi-metric property for every vertices u, w, v and x: if a shortest path between u and w and a shortest path between x and v share a terminal edge vw,Roughly, gluing together any two shortest paths along a common terminal edge may not necessarily result in a shortest path but yields a "near-shortest" path with defect at most i. It is known that α0-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are αi-metric for i = 1 and i = 2, respectively. We show that an additive O(i)-approximation of the radius, of the diameter, and in fact of all vertex eccentricities of an αi-metric graph can be computed in total linear time. Our strongest results are obtained for α1-metric graphs, for which we prove that a central vertex can be computed in subquadratic time, and even better in linear time for so-called (α1, ∆)-metric graphs (a superclass of chordal graphs and of plane triangulations with inner vertices of degree at least 7). The latter answers a question raised in (Dragan, IPL, 2020). Our algorithms follow from new results on centers and metric intervals of αi-metric graphs. In particular, we prove that the diameter of the center is at most 3i + 2 (at most 3, if i = 1). The latter partly answers a question raised in (Yushmanov & Chepoi, Mathematical Problems in Cybernetics, 1991).

Injective hulls of various graph classes

“…Helly graphs have been well-investigated; they have several characterizations and important features as established in [4,5,17,18,20,33,35]. They are exactly the so-called absolute retracts of reflexive graphs and possess a certain elimination scheme [4,5,17,18,33] which makes them recognizable in O(n 2 m) time [17].…”

“…Conveniently, the eccentricity function in Helly graphs is unimodal [18], that is, any local minimum coincides with the global minimum. This fact was recently used in [19,20,27] to compute the radius, diameter and a central vertex of a Helly graph in subquadratic time. Helly graphs can be metrically characterized by the fact that all disks of uniform radius have the Helly property [20].…”

Distance problems within Helly graphs and $k$-Helly graphs

“…We stress that in [36], we used different properties of unimodal functions in order to compute a diametral pair. In an upcoming paper [29], based on the techniques developed here and in [36], we prove that we can also compute the center of a Helly graph in truly subquadratic time (i.e., the set of all its central vertices). Note that for a Helly graph, computing the center is equivalent to computing all the eccentricities [33].…”

Beyond Helly graphs: the diameter problem on absolute retracts

“…A function is called unimodal if every local minimum is also a global minimum. It is known that the eccentricity function of a Helly graph is unimodal [36], and this property got used in [40] in order to compute all the eccentricities in this graph class in subquadratic time. Next, we prove that a similar, but weaker property holds for each colour class of absolute retracts, namely: Lemma 24.…”

“…Then, we show how to compute the diameter of G from the diameter of both Helly graphs (actually, from the knowledge of the peripheral vertices in these graphs, i.e., those vertices with maximal eccentricity). Recently [40], we announced an O(m √ n)-time algorithm in order to compute all the eccentricities in a Helly graph. However, extending this result to the absolute retracts of bipartite graphs appears to be a more challenging task.…”