Feodor F. Dragan

Kent State University | KSU · Department of Computer Science

A ($\lambda,\mu$)-bow metric was defined in (Dragan & Ducoffe, 2023) as a far reaching generalization of an $\alpha_i$-metric (which is equivalent to a ($0,i$)-bow metric). A graph $G=(V,E)$ is said to satisfy ($\lambda,\mu$)-bow metric if for every four vertices $u,v,w,x$ of $G$ the following holds: if two shortest paths $P(u,w)$ and $P(v,x)$ shar...

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 $$\alpha _i$$ α i -metric ( $$i\in {\mathcal {N}}$$ i ∈ N ) if it satisfies the following $$\alpha _i$$ α i -metric property for every vertices u , w , v and x...

This Poster presents a new method of analyzing and classifying liquid crystal textures, using feed-forward neural networks and different clustering techniques. Liquid crystal phases are generally identified by human experts by polarizing optical microscopy observations of textures, based on typical defects, the smoothness or sharpness of domains, a...

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 \(\alpha _i\)-metric (\(i\in \mathcal {N}\)) if it satisfies the following \(\alpha _i\)-metric property for every vertices u, w, v and x: if a shortest path be...

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 $\alpha_i$-metric ($i\in \mathcal{N}$) if it satisfies the following $\alpha_i$-metric property for every vertices $u,w,v$ and $x$: if a shortest path between $...

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 \(\mathcal {H}(G)\) into which G isometrically embeds; \(\mathcal {H}(G)\) is called the injective hull of G. Motivated by this, we...

We investigate a metric parameter “Leanness” of graphs which is a formalization of a well-known Fellow Travelers Property present in some metric spaces. Given a graph \(G=(V,E)\), the leanness of G is the smallest \(\lambda \) such that, for every pair of vertices \(x,y\in V\), all shortest (x, y)-paths stay within distance \(\lambda \) from each o...

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 d...

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. Our main contribution in this note is a new characterization of hyperbolicit...

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)....

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 d...

We present new algorithmic results for the class of Helly graphs, that is, for the discrete analogues of hyperconvex metric spaces. Specifically, an undirected unweighted graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. It is known that every graph isometrically embeds into a Helly graph that makes o...

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 pr...

A new metric parameter for a graph, Helly-gap, is introduced. A graph $G$ is called $\alpha$-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 $\alpha$. The minimum $\alpha$ for which a graph $G$ is $\alpha$-weakly-Helly is called the He...

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∈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∈V} is the radius of gra...

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), d(u,x)+d(v,w) differ by at most 2δ≥0. This paper describes the eccentricity terrain of a δ-hyperbolic graph. The eccentricity function eG(v)=max⁡{d(v,u):u∈V} partitions vertices of G into eccentricity layers Ck(G...

A graph $G=(V,E)$ is $\delta$-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 $2\delta \geq 0$. Recent work shows that many real-world graphs have small hyperbolicity $\delta$. This paper describes the eccentricity terrain of a $\delta$-...

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 (with the former being a subclass of Helly graphs and the latter being a particular case of graphs of bounded fractional Helly number, respectively) we pr...

It is known that every chordal graph G=(V,E) has a spanning tree T such that, for every vertex v∈V, eccT(v)≤eccG(v)+2 holds (here eccG(v):=max⁡{dG(v,u):u∈V} is the eccentricity of v in G). We show that such a spanning tree can be computed in linear time for every chordal graph. As a byproduct, we get that the eccentricities of all vertices of a cho...

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) = \m...

In this paper, we study how the Pruned Landmark Labeling (PPL) algorithm can be parallelized in a scalable fashion, producing the same results as the sequential algorithm. More specifically, we parallelize using a Vertex-Centric (VC) computational model on a modern SIMD powered multicore architecture. We design a new VC-PLL algorithm that resolves...

We show that the eccentricities of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, 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):=\max\{d_G(u,v): u\i...

We show that the eccentricities (and thus the centrality indices) of all vertices of a -hyperbolic graph can be computed in linear time with an additive one-sided error of at most , i.e., after a linear time preprocessing, for every vertex v of G one can compute in O(1) time an estimate of its eccentricity such that for a small constant c. We prove...

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, 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 eccentric...

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 hyperbo...

We introduce notions of certificates allowing to bound eccentricities in a graph. In particular , we revisit radius (minimum eccentricity) and diameter (maximum eccentricity) computation and explain the efficiency of practical radius and diameter algorithms by the existence of small certificates for radius and diameter plus few additional propertie...

The [Formula presented]-hyperbolicity of a graph is defined by a simple 4-point condition: for any four vertices [Formula presented], [Formula presented], [Formula presented], and [Formula presented], the two larger of the distance sums [Formula presented], [Formula presented], and [Formula presented] differ by at most [Formula presented]. Hyperbol...

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 hyperb...

Using the characteristic property of chordal graphs that they are the intersection graphs of subtrees of a tree, Erich Prisner showed that every chordal graph admits an eccentricity 2-approximating spanning tree. That is, every chordal graph G has a spanning tree T such that eccT(v)-eccG(v)≤2 for every vertex v, where eccG(v) (eccT(v)) is the eccen...

In this paper, we introduce and investigate the Minimum Eccentricity Shortest Path (MESP) problem in unweighted graphs. It asks for a given graph to find a shortest path with minimum eccentricity. Let n and m denote the number of vertices and the number of edges of a given graph. We demonstrate that: • a minimum eccentricity shortest path plays a c...

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) $\big(r + \mathcal{O}(\mu) \big)$-dominating set $D$ with $|D| \leq |D^*|$ efficiently. He...

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) $\big(r + \mathcal{O}(\mu) \big)$-dominating set $D$ with $|D| \leq |D^*|$ efficiently. He...

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 $\delta$-slim if for any verte...

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 $\delta$-slim if for any verte...

For a graph the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices x, y the distance on the line can be bounded by the term , where is the distance in the graph. The minimum bandwidth problem minimizes the term , where f is a mapping of th...

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network G admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and...

In this paper, we introduce and investigate a new notion of strong tree-breadth. We say that a graph G has strong tree-breadth \(\rho \) if there is a tree-decomposition T for G such that each bag B of T is equal to the complete \(\rho \)-neighbourhood of some vertex v in G, i. e., \(B = N_G^\rho [v]\). We show that it is NP-complete to determine i...

Hyperbolicity is a global property of graphs that measures how close their structures are to trees in terms of their distances. It embeds multiple properties that facilitate solving several problems that found to be hard in the general graph form. In this paper, we investigate the hyperbolicity of graphs not only by considering Gromov’s notion of δ...

We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a minimum eccentricity shortest path can be found in linear time for distance-hereditary graphs (generalizing the p...

Using the characteristic property of chordal graphs that they are the intersection graphs of subtrees of a tree, Erich Prisner showed that every chordal graph admits an eccentricity 2-approximating spanning tree. That is, every chordal graph G has a spanning tree T such that \(ecc_T(v)-ecc_G(v)\le 2\) for every vertex v, where \(ecc_G(v)\) (\(ecc_T...

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan an...

Recent advances in the field of genetic data analysis reveal promising findings in the field of human history; especially when combined with proper data analysis tools. Within the field of modern genetics, there is evidence that the human populations have genetically interacted as a result of several events. The genetic admixture contains multiple...

We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a minimum eccentricity shortest path can be found in linear time for distance-hereditary graphs (generalizing the p...

We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a minimum eccentricity shortest path can be found in linear time for distance-hereditary graphs (generalizing the p...

In this paper, we introduce and investigate the Minimum Eccentricity Shortest Path (MESP) problem in unweighted graphs. It asks for a given graph to find a shortest path with minimum eccentricity. We demonstrate that: a minimum eccentricity shortest path plays a crucial role in obtaining the best to date approximation algorithm for a minimum distor...

Recently in few papers, a balanced disk separator of graphs were recursively used to construct one or a small set of spanning trees that sharply approximate distances in a given graph. The best up to date approximation algorithms were obtained for multiplicative tree spanners and for collective additive tree spanners of graphs. In this thesis, we a...

Hyperbolicity is a global property of graphs that measures how close their structures are to trees in terms of their distances. It embeds multiple properties that facilitate solving several problems that found to be hard in the general graph form. In this paper, we investigate the hyperbolicity of graphs not only by considering Gromov’s notion of δ...

For a graph \(G=(V,E)\) the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices x, y the distance on the line \(|f(x) - f(y)|\) can be bounded by the term \(d_G(x, y)\le |f(x)-f(y)|\le k \, d_G(x, y)\), where \(d_G(x, y)\) is the distance i...

We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its...

Based on solid theoretical foundations, we present strong evidences that a number of real-life networks, taken from different domains like Internet measurements, biological datasets, web graphs, social and collaboration networks, exhibit tree-like structures from a metric point of view. We investigate few graph parameters, namely, the tree-distorti...

Recent empirical and theoretical work has suggested that many real-life complex networks and graphs arising in Internet applications, in biological and social sciences, in chemistry and physics have tree-like structures from a metric point of view. A number of graph parameters trying to capture this phenomenon and to measure these tree-like structu...

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph $G$ {\em admits a system of $\mu$ collective additive tree $r$-spanners} (resp., {\em multiplicative tree $t$-spanners}) if there...

δ-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points u,v,w,x, the two larger of the distance sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of in...

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable...

Transactional data are ubiquitous. Several methods, including frequent itemset mining and co-clustering, have been proposed to analyze transactional databases. In this work, we propose a new research problem to succinctly summarize transactional databases. Solving this problem requires linking the high level structure of the database to a potential...

A t-spanner of a graph G is its spanning subgraph S such that the distance between every pair of vertices in S is at most t times their distance in G. The sparsestt-spanner problem asks to find, for a given graph G and an integer t, a t-spanner of G with the minimum number of edges. The problem is known to be NP-hard for all t≥2, and, even more, it...

Let G=(V,E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z=x), unless z=y, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighbor...

A spanning tree T of a graph G is called a tree t-spanner of G if the distance between every pair of vertices in T is at most t times their distance in G. In this paper, we present an algorithm which constructs for an n-vertex m-edge unweighted graph G: (1) a tree (2ëlog2 nû)(2\lfloor\log_2 n\rfloor)-spanner in O(mlogn) time, if G is a chordal grap...

In this article, motivated by applications of ordinary (distance) spanners in communication networks and to address such issues as bandwidth constraints on network links, link failures, network survivability, etc., we introduce a new notion of flow spanner, where one seeks a spanning subgraph H = (V, E') of a graph G = (V, E) which provides a “good...

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu et al. (Proceedings of the eighteenth annual ACM–SIAM symposium on discrete algorithms (SODA 2007), pp. 512–521, 2007)...

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system...

A distance-k matching in a graph G is matching M in which the distance between any two edges of M is at least k. A distance-2 matching is more commonly referred to as an induced matching. In this paper, we show that when G is weakly chordal, the size of the largest induced matching in G is equal to the minimum number of co-chordal subgraphs of G ne...

Various network monitoring and performance evaluation schemes generate considerable amount of traf- fic, which affects network performance. In this paper we describe a method for minimizing network monitoring overhead based on Shortest Path Tree (SPT) protocol. We describe two different variations of the problem: the AProblem and the E-Problem and...

In this paper, we investigate three strategies of how to use a spanning tree T of a graph G to navigate in G, i.e., to move from a current vertex x towards a destination vertex y via a path that is close to optimal. In each strategy, each vertex v has full knowledge of its neighborhood N G [v] in G (or, k-neighborhood D k (v,G), where k is a small...

In this paper, we propose a new compact and low delay routing labeling scheme for Unit Disk Graphs (UDGs) which often model wireless ad hoc networks. We show that one can assign each vertex of an n–vertex UDG G a compact O(log2 n)-bit label such that, given the label of a source vertex and the label of a destination, it is possible to compute effic...

In this paper, we establish a novel balanced separator theorem for Unit Disk Graphs (UDGs), which mimics the well-known Lipton and Tarjan's planar balanced shortest paths separator theorem. We prove that, in any n-vertex UDG G, one can find two hop-shortest paths P(s,x) and P(s,y) such that the removal of the 3-hop-neighborhood of these paths (i.e....

In this paper, we investigate three strategies of how to use a spanning tree T of a graph G to navigate in G, i.e., to move from a current vertex x towards a destination vertex y via a path that is close to optimal. In each strategy, each vertex v has full knowledge of its neighborhood N G [v] in G (or, k-neighborhood D k (v,G), where k is a smal...

Let G = (V,E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its ne...

In this work, we study a visual data mining problem: Given a set of discovered overlapping submatrices of inter- est, how can we order the rows and columns of the data matrix to best display these submatrices and their relation- ships? We find this problem can be converted to the hyper- graph ordering problem, which generalizes the traditional mini...

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. The sparsest t -spanner problem asks to find, for a given graph G and an integer t, a t-spanner of G with the minimum number of edges. On general n-vertex graphs, the problem is known to be NP-hard for all...

Transactional data are ubiquitous. Several methods, including frequent itemsets mining and co-clustering, have been proposed to analyze transactional databases. In this work, we propose a new research problem to succinctly summarize transactional databases. Solving this problem requires linking the high level structure of the database to a potentia...

delta-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points u, v, w, x, the two larger of the sums d(u, v) + d(w, x), d(u, w) + d(v, x), d(u, x) + d(v, w) differ by at most 2 delta. Given a finite set S of points of a delta-hyperbolic space, we present simple and fast methods for approximating t...

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t -spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs the tree t -spanner problem (the problem to decide whether G admits a...

A graph G = (V,E) is said to admit a system of μ collective additive tree r-spanners if there is a system T\cal{T}(G) of at most μ spanning trees of G such that for any two vertices u,v of G a spanning tree T Î TT\in \cal{T}(G) exists such that the distance in T between u and v is at most r plus their distance in G. In this paper, we examine the pr...