arXiv subjects

Feodor F. Dragan

Publications and source records attributed to Feodor F. Dragan.

At least 19 recordsLinked to original sources

arXiv2026

$α_i$-Metric Graphs: Hyperbolicity

A graph is called $α_i$-metric ($i \in {\cal N}$) if it satisfies the following $α_i$-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 $vw$, then $d(u,x) \ge d(u,v) + d(v,x) - i$. The latter is a discrete relaxation of the property that in Euclidean spaces the union of two geodesics sharing a terminal segment must be also a geodesic. Recently in (Dragan & Ducoffe, WG'23) we initiated the study of the algorithmic applications of $α_i$-metric graphs. Our results in this prior work were very similar to those established in (Chepoi et al., SoCG'08) and (Chepoi et al., COCOA'18) for graphs with bounded hyperbolicity. The latter is a heavily studied metric tree-likeness parameter first introduced by Gromov. In this paper, we clarify the relationship between hyperbolicity and the $α_i$-metric property, proving that $α_i$-metric graphs are $f(i)$-hyperbolic for some function $f$ linear in $i$. We give different proofs of this result, using various equivalent definitions to graph hyperbolicity. By contrast, we give simple constructions of $1$-hyperbolic graphs that are not $α_i$-metric for any constant $i$. Finally, in the special case of $i=1$, we prove that $α_1$-metric graphs are $1$-hyperbolic, and the bound is sharp. By doing so, we can answer some questions left open in (Dragan & Ducoffe, WG'23).

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arXiv2026

Certificates in P and Subquadratic-Time Computation of Radius, Diameter, and all Eccentricities in Graphs

In the context of fine-grained complexity, we investigate the notion of certificate enabling faster polynomial-time algorithms. We specifically target radius (minimum eccentricity), diameter (maximum eccentricity), and all-eccentricity computations for which quadratic-time lower bounds are known under plausible conjectures. In each case, we introduce a notion of certificate as a specific set of nodes from which appropriate bounds on all eccentricities can be derived in subquadratic time when this set has sublinear size. The existence of small certificates for radius, diameter and all eccentricities is a barrier against SETH-based lower bounds for these problems. We indeed prove that for graph classes with certificates of bounded size, there exist randomized subquadratic-time algorithms for computing the radius, the diameter, and all eccentricities respectively. Moreover, these notions of certificates are tightly related to algorithms probing the graph through one-to-all distance queries and allow to explain the efficiency of practical radius and diameter algorithms from the literature. In particular, our formalization enables a novel primal-dual analysis of a classical approach for diameter computation. Based on our novel insights for these problems, we introduce several new algorithmic techniques related to eccentricity computation and propose algorithms for radius, diameter and all eccentricities with theoretical guarantees with respect to certain graph parameters. This is complemented by experimental results on various types of real-world graphs showing that these parameters appear to be low in practice. Finally, we obtain refined results in the case where the input graph is a power-law random graph, has low doubling dimension, has low hyperbolicity, is chordal, satisfies some Helly-type property, or has bounded asteroidal number.

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arXiv2025

Lower bounds on collective additive spanners

In this paper we present various lower bound results on collective tree spanners and on spanners of bounded treewidth. A graph $G$ is said to admit a system of $μ$ collective additive tree $c$-spanners if there is a system $\cal{T}$$(G)$ of at most $μ$ spanning trees of $G$ such that for any two vertices $u,v$ of $G$ a tree $T\in \cal{T}$$(G)$ exists such that the distance in $T$ between $u$ and $v$ is at most $c$ plus their distance in $G$. A graph $G$ is said to admit an additive $k$-treewidth $c$-spanner if there is a spanning subgraph $H$ of $G$ with treewidth $k$ such that for any pair of vertices $u$ and $v$ their distance in $H$ is at most $c$ plus their distance in $G$. Among other results, we show that: $\bullet$ Any system of collective additive tree $1$ -- spanners must have $Ω(\sqrt[3]{\log n})$ spanning trees for some unit interval graphs; $\bullet$ No system of a constant number of collective additive tree $2$-spanners can exist for strongly chordal graphs; $\bullet$ No system of a constant number of collective additive tree $3$-spanners can exist for chordal graphs; $\bullet$ No system of a constant number of collective additive tree $c$-spanners can exist for weakly chordal graphs as well as for outerplanar graphs for any constant $c\geq 0$; $\bullet$ For any constants $k \ge 2$ and $c \ge 1$ there are graphs of treewidth $k$ such that no spanning subgraph of treewidth $k-1$ can be an additive $c$-spanner of such a graph. All these lower bound results apply also to general graphs. Furthermore, they %results complement known upper bound results with tight lower bound results.

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arXiv2025

Graph parameters that are coarsely equivalent to path-length

Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length of a tree-decomposition ${\cal T}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal T}(G)$, and the tree-length $tl(G)$ of $G$ is the minimum of the length, over all tree-decompositions of $G$. Similarly, the length of a path-decomposition ${\cal P}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal P}(G)$, and the path-length $pl(G)$ of $G$ is the minimum of the length, over all path-decompositions of $G$. In this paper, we present several graph parameters that are coarsely equivalent to path-length. Among other results, we show that the path-length of a graph $G$ is small if and only if one of the following equivalent conditions is true: (a) $G$ can be embedded to an unweighted caterpillar tree (equivalently, to a graph of path-width one) with a small additive distortion; (b) there is a constant $r\ge 0$ such that for every triple of vertices $u,v,w$ of $G$, disk of radius $r$ centered at one of them intercepts all paths connecting two others; (c) $G$ has a $k$-dominating shortest path with small $k\ge 0$; (d) $G$ has a $k'$-dominating pair with small $k'\ge 0$; (e) some power $G^μ$ of $G$ is an AT-free (or even a cocomparability) graph for a small integer $μ\ge 0$.

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arXiv2025

Graph parameters that are coarsely equivalent to tree-length

Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length of a tree-decomposition ${\cal T}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal T}(G)$, and the tree-length of $G$ is the minimum of the length, over all tree-decompositions of $G$. We present simpler and sometimes with better bounds proofs for those known in literature results and further extend this list of graph parameters coarsely equivalent to tree-length. Among other new results, we show that the tree-length of a graph $G$ is small if and only if for every bramble ${\cal F}$ (or every Helly family of connected subgraphs ${\cal F}$, or every Helly family of paths ${\cal F}$) of $G$, there is a disk in $G$ with small radius that intercepts all members of ${\cal F}$. Furthermore, the tree-length of a graph $G$ is small if and only if $G$ can be embedded with a small additive distortion to an unweighted tree with the same vertex set as in $G$ (not involving any Steiner points). Additionally, we introduce a new natural "bridging`` property for cycles, which generalizes a known property of cycles in chordal graphs, and show that it also coarsely defines the tree-length.

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arXiv2024

Bow Metrics and Hyperbolicity

A ($λ,μ$)-bow metric was defined in (Dragan & Ducoffe, 2023) as a far reaching generalization of an $α_i$-metric (which is equivalent to a ($0,i$)-bow metric). A graph $G=(V,E)$ is said to satisfy ($λ,μ$)-bow metric if for every four vertices $u,v,w,x$ of $G$ the following holds: if two shortest paths $P(u,w)$ and $P(v,x)$ share a common shortest subpath $P(v,w)$ of length more than $λ$ (that is, they overlap by more than $λ$), then the distance between $u$ and $x$ is at least $d_G(u,v)+d_G(v,w)+d_G(w,x)-μ$. ($λ,μ$)-Bow metric can also be considered for all geodesic metric spaces. It was shown by Dragan & Ducoffe that every $δ$-hyperbolic graph (in fact, every $δ$-hyperbolic geodesic metric space) satisfies ($δ, 2δ$)-bow metric. Thus, ($λ,μ$)-bow metric is a common generalization of hyperbolicity and of $α_i$-metric. In this paper, we investigate an intriguing question whether ($λ,μ$)-bow metric implies hyperbolicity in graphs. Note that, this is not the case for general geodesic metric spaces as Euclidean spaces satisfy ($0,0$)-bow metric whereas they have unbounded hyperbolicity. We conjecture that, in graphs, ($λ,μ$)-bow metric indeed implies hyperbolicity and show that our conjecture is true for several large families of graphs.

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arXiv2023

$α_i$-Metric Graphs: Radius, Diameter and all Eccentricities

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 $α_i$-metric ($i\in \mathcal{N}$) if it satisfies the following $α_i$-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 $vw$, then $d(u,x)\geq d(u,v) + d(v,x)-i$. 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 $α_0$-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are $α_i$-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 $α_i$-metric graph can be computed in total linear time. Our strongest results are obtained for $α_1$-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 $(α_1,Δ)$-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 $α_i$-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).

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arXiv2021

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

A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. The class of Helly graphs is the discrete analogue of the class of hyperconvex metric spaces. It is also known that every graph isometrically embeds into a Helly graph, making the latter an important class of graphs in Metric Graph Theory. We study diameter, radius and all eccentricity computations within the Helly graphs. Under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, it was recently shown that the radius and the diameter of an $n$-vertex $m$-edge Helly graph $G$ can be computed with high probability in $\tilde{\mathcal O}(m\sqrt{n})$ time (i.e., subquadratic in $n+m$). In this paper, we improve that result by presenting a deterministic ${\mathcal O}(m\sqrt{n})$ time algorithm which computes not only the radius and the diameter but also all vertex eccentricities in a Helly graph. Furthermore, we give a parameterized linear-time algorithm for this problem on Helly graphs, with the parameter being the Gromov hyperbolicity $δ$. More specifically, we show that the radius and a central vertex of an $m$-edge $δ$-hyperbolic Helly graph $G$ can be computed in $\mathcal O(δm)$ time and that all vertex eccentricities in $G$ can be computed in $\mathcal O(δ^2 m)$ time. To show this more general result, we heavily use our new structural properties obtained for Helly graphs.

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arXiv2020

Eccentricity function in distance-hereditary graphs

A graph $G=(V,E)$ is distance hereditary if every induced path of $G$ is a shortest path. In this paper, we show that the eccentricity function $e(v)=\max\{d(v,u): u\in V\}$ in any distance-hereditary graph $G$ is almost unimodal, that is, every vertex $v$ with $e(v)> rad(G)+1$ has a neighbor with smaller eccentricity. Here, $rad(G)=\min\{e(v): v\in V\}$ is the radius of graph $G$. Moreover, we use this result to fully characterize the centers of distance-hereditary graphs. Several bounds on the eccentricity of a vertex with respect to its distance to the center of $G$ or to the ends of a diametral path are established. Finally, we propose a new linear time algorithm to compute all eccentricities in a distance-hereditary graph.

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arXiv2020

Injective hulls of various graph classes

A graph 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 a unique smallest Helly graph H(G) into which G isometrically embeds; H(G) is called the injective hull of G. Motivated by this, we investigate the structural properties of the injective hulls of various graph classes. We say that a class of graphs $\mathcal{C}$ is closed under Hellification if $G \in \mathcal{C}$ implies $H(G) \in \mathcal{C}$. We identify several graph classes that are closed under Hellification. We show that permutation graphs are not closed under Hellification, but chordal graphs, square-chordal graphs, and distance-hereditary graphs are. Graphs that have an efficiently computable injective hull are of particular interest. A linear-time algorithm to construct the injective hull of any distance-hereditary graph is provided and we show that the injective hull of several graphs from some other well-known classes of graphs are impossible to compute in subexponential time. In particular, there are split graphs, cocomparability graphs, bipartite graphs G such that H(G) contains $Ω(a^{n})$ vertices, where $n=|V(G)|$ and $a>1$.

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arXiv2020

Eccentricity terrain of $δ$-hyperbolic graphs

A graph $G=(V,E)$ is $δ$-hyperbolic if for any four vertices $u,v,w,x$, the two larger of the three 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 $2δ\geq 0$. Recent work shows that many real-world graphs have small hyperbolicity $δ$. This paper describes the eccentricity terrain of a $δ$-hyperbolic graph. The eccentricity function $e_G(v)=\max\{d(v,u) : u \in V\}$ partitions the vertex set of $G$ into eccentricity layers $C_{k}(G) = \{v \in V : e(v)=rad(G)+k\}$, $k \in \mathbb{N}$, where $rad(G)=\min\{e_G(v): v\in V\}$ is the radius of $G$. The paper studies the eccentricity layers of vertices along shortest paths, identifying such terrain features as hills, plains, valleys, terraces, and plateaus. It introduces the notion of $β$-pseudoconvexity, which implies Gromov's $ε$-quasiconvexity, and illustrates the abundance of pseudoconvex sets in $δ$-hyperbolic graphs. In particular, it shows that all sets $C_{\leq k}(G)=\{v\in V : e_G(v) \leq rad(G) + k\}$, $k\in \mathbb{N}$, are $(2δ-1)$-pseudoconvex. Additionally, several bounds on the eccentricity of a vertex are obtained which yield a few approaches to efficiently approximating all eccentricities. An $O(δ|E|)$ time eccentricity approximation $\hat{e}(v)$, for all $v\in V$, is presented that uses distances to two mutually distant vertices and satisfies $e_G(v)-2δ\leq \hat{e}(v) \leq {e_G}(v)$. It also shows existence of two eccentricity approximating spanning trees $T$, one constructible in $O(δ|E|)$ time and the other in $O(|E|)$ time, which satisfy ${e}_G(v) \leq e_T(v) \leq {e}_G(v)+4δ+1$ and ${e}_G(v) \leq e_T(v) \leq {e}_G(v)+6δ$, respectively. Thus, the eccentricity terrain of a tree gives a good approximation (up-to an additive error $O(δ))$ of the eccentricity terrain of a $δ$-hyperbolic graph.

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arXiv2020

Helly-gap of a graph and vertex eccentricities

A new metric parameter for a graph, Helly-gap, is introduced. A graph $G$ is called $α$-weakly-Helly if any system of pairwise intersecting disks in $G$ has a nonempty common intersection when the radius of each disk is increased by an additive value $α$. The minimum $α$ for which a graph $G$ is $α$-weakly-Helly is called the Helly-gap of $G$ and denoted by $α(G)$. The Helly-gap of a graph $G$ is characterized by distances in the injective hull $\mathcal{H}(G)$, which is a (unique) minimal Helly graph which contains $G$ as an isometric subgraph. This characterization is used as a tool to generalize many eccentricity related results known for Helly graphs ($α(G)=0$), as well as for chordal graphs ($α(G)\le 1$), distance-hereditary graphs ($α(G)\le 1$) and $δ$-hyperbolic graphs ($α(G)\le 2δ$), to all graphs, parameterized by their Helly-gap $α(G)$. Several additional graph classes are shown to have a bounded Helly-gap, including AT-free graphs and graphs with bounded tree-length, bounded chordality or bounded $α_i$-metric.

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arXiv2019

A story of diameter, radius and Helly property

A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. Motivated by previous work on dually chordal graphs and graphs of bounded distance VC-dimension we prove several new results on the complexity of computing the diameter and the radius on Helly graphs and related graph classes. * First, we present algorithms which given an $n$-vertex $m$-edge Helly graph $G$ as input, compute w.h.p. its radius and its diameter in time $\tilde{\cal O}(m\sqrt{n})$. Our algorithms are based on the Helly property and on several implications of the unimodality of the eccentricity function in Helly graphs: every vertex of locally minimum eccentricity is a central vertex. * Then, we focus on $C_4$-free Helly graphs, which include, amongst other subclasses, bridged Helly graphs and so, chordal Helly graphs and hereditary Helly graphs. For the $C_4$-free Helly graphs, we present linear-time algorithms for computing the eccentricity of all vertices. Doing so, we generalize previous results on strongly chordal graphs to a much larger subclass. * Finally, we derive from our findings on chordal Helly graphs a more general one-to-many reduction from diameter computation on chordal graphs to either diameter computation on split graphs or the {\sc Disjoint Set} problem. Therefore, split graphs are in some sense the {\em only} hard instances for diameter computation on chordal graphs. As a byproduct of our reduction the eccentricity of all vertices in a chordal graph can be approximated in ${\cal O}(m\log{n})$ time with an additive one-sided error of at most one, and on any subclass of chordal graphs with constant VC-dimension the diameter can be computed in truly subquadratic time. These above results are a new step toward better understanding the role of abstract geometric properties in the fast computation of metric graph invariants.

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arXiv2019

Obstructions to a small hyperbolicity in Helly graphs

It is known that for every graph $G$ there exists the smallest Helly graph $\cal H(G)$ into which $G$ isometrically embeds ($\cal H(G)$ is called the injective hull of $G$) such that the hyperbolicity of $\cal H(G)$ 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.

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arXiv2019

Fast approximation and exact computation of negative curvature parameters of graphs

In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomial-time, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm for computing the hyperbolicity of an $n$-vertex graph $G=(V,E)$ in optimal time $O(n^2)$ (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space. We also show that a similar characterization of hyperbolicity holds for all geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we prove that any complete geodesic metric space $(X,d)$ has such a geodesic spanning tree. We hope that this fundamental result can be useful in other contexts.

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arXiv2019

Slimness of graphs

Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph $G=(V,E)$, a geodesic triangle $\bigtriangleup(x,y,z)$ with $x, y, z\in V$ is the union $P(x,y) \cup P(x,z) \cup P(y,z)$ of three shortest paths connecting these vertices. A geodesic triangle $\bigtriangleup(x,y,z)$ is called $δ$-slim if for any vertex $u\in V$ on any side $P(x,y)$ the distance from $u$ to $P(x,z) \cup P(y,z)$ is at most $δ$, i.e. each path is contained in the union of the $δ$-neighborhoods of two others. A graph $G$ is called $δ$-slim, if all geodesic triangles in $G$ are $δ$-slim. The smallest value $δ$ for which $G$ is $δ$-slim is called the slimness of $G$. In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter $Δ(G)$ of a layering partition of $G$, (2) graphs with tree-length $λ$, (3) graphs with tree-breadth $ρ$, (4) $k$-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show that the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraph is at most 1.

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arXiv2018

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $δ$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $cδ$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ cδ$ for a small constant $c$. We prove that every $δ$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ cδ$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'δ$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

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arXiv2017

Parameterized Approximation Algorithms for some Location Problems in Graphs

We develop efficient parameterized, with additive error, approximation algorithms for the (Connected) $r$-Domination problem and the (Connected) $p$-Center problem for unweighted and undirected graphs. Given a graph $G$, we show how to construct a (connected) $\big(r + \mathcal{O}(μ) \big)$-dominating set $D$ with $|D| \leq |D^*|$ efficiently. Here, $D^*$ is a minimum (connected) $r$-dominating set of $G$ and $μ$ is our graph parameter, which is the tree-breadth or the cluster diameter in a layering partition of $G$. Additionally, we show that a $+ \mathcal{O}(μ)$-approximation for the (Connected) $p$-Center problem on $G$ can be computed in polynomial time. Our interest in these parameters stems from the fact that in many real-world networks, including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others, and in many structured classes of graphs these parameters are small constants.

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