Victor Chepoi

Distance Approximating Trees for Chordal and Dually Chordal Graphs

In this paper we show that, for each chordal graphG, there is a treeTsuch thatTis a spanning tree of the squareG2ofGand, for every two vertices, the distance between them inTis not larger than the distance inGplus 2. Moreover, we prove that, ifGis a strongly chordal graph or even a dually chordal graph, then there exists a spanning treeTofGthat is an additive 3-spanner as well as a multiplicative 4-spanner ofG. In all cases the treeTcan be computed in linear time.

The Wiener Index and the Szeged Index of Benzenoid Systems in Linear Time

A linear time algorithm is presented which, for a given benzenoid system G, computes the Wiener index of G. The algorithm is based on an isometric embedding of G into the Cartesian product of three trees, combined with the notion of the Wiener index of vertex-weighted graphs. An analogous approach yields also a linear algorithm for computing the Szeged index of benzenoid systems.

The algorithmic use of hypertree structure and maximum neighbourhood orderings

The use of (generalized) tree structure in graphs is one of the main topics in the field of efficient graph algorithms. The well-known partial κ-tree (resp. treewidth) approach belongs to this kind of research and bases on a tree structure of constant-size bounded maximal cliques. Without size bound on the cliques this tree structure of maximal cliques characterizes chordal graphs which are known to be important also in connection with relational database schemes where hypergraphs with tree structure (acyclic hypergraphs) and their elimination orderings (perfect elimination orderings for chordal graphs, Graham-reduction for acyclic hypergraphs) are studied.

Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points <i>u,v,w,x</i>, the two larger of the sums <i>d</i>(<i>u,v</i>)+<i>d</i>(<i>w,x</i>), <i>d</i>(<i>u,w</i>)+<i>d</i>(<i>v,x</i>), <i>d</i>(<i>u,x</i>)+<i>d</i>(<i>v,w</i>) differ by at most 2δ. Given a finite set <i>S</i> of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of <i>S</i> with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for <i>S</i> with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs <i>G</i>=(<i>V,E</i>) with uniformly bounded degrees of vertices, the exact center of <i>S</i> can be computed in linear time <i>O</i>(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs <i>G</i> on <i>n</i> vertices with an additive error <i>O</i>(δlog₂ <i>n</i>). This construction has an additive error comparable with that given by Gromov for <i>n</i>-point δ-hyperbolic spaces, but can be implemented in <i>O</i>(|E|) time (instead of <i>O</i>(<i>n</i>²)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.

Clin d'oeil on L 1 -embeddable planar graphs

In this note we present some properties of LI-embeddable planar graphs. We present a characterization of graphs isometrically embeddable into half-cubes. This result implies that every planar Li-graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of a planar Li-graph G the subgraph H of G bounded by C is also Li-embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the LI-embeddability of a list of planar graphs.

Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs

We present simple methods for approximating the diameters, radii, and centers of finite sets in δ-hyperbolic geodesic spaces and graphs. We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δ log 2 n) comparable with that given by M. Gromov.

A Note on Distance Approximating Trees in Graphs

Let G= (V, E) be a connected graph endowed with the standard graph-metric dGand in which longest induced simple cycle has length? (G). We prove that there exists a tree T= (V,F ) such that| dG(u, v) ?dT(u, v)| ? ??(G)2? +?for all vertices u, v?V, where?= 1 if ?(G) ?= 4, 5 and ?= 2 otherwise. The case ?(G) = 3 (i.e., G is a chordal graph) has been considered in Brandstadt, Chepoi, and Dragan, (1999) J.Algorithms 30. The proof contains an efficient algorithm for determining such a treeT .

Packing and Covering δ-Hyperbolic Spaces by Balls

We consider the problem of covering and packing subsets ofΔ-hyperbolic metric spaces and graphs by balls.These spaces, defined via a combinatorial Gromov condition, haverecently become of interest in several domains of computer science.Specifically, given a subset Sof aΔ-hyperbolic graph Gand a positive numberR, let Δ(S,R) be theminimum number of balls of radius Rcovering S.It is known that computing Δ(S,R)or approximating this number within a constant factor is hard evenfor 2-hyperbolic graphs. In this paper, using a primal-dualapproach, we show how to construct in polynomial time a covering ofSwith at most Δ(S,R)balls of (slightly larger) radius R+ Δ.This result is established in the general framework ofΔ-hyperbolic geodesic metric spaces and is extendedto some other set families derived from balls. The coveringalgorithm is used to design better approximation algorithms for theaugmentation problem with diameter constraints and for thek-center problem in Δ-hyperbolicgraphs.

On Distances in Benzenoid Systems

We show that the molecular graph G of a benzenoid hydrocarbon admits an isometric embedding into the Cartesian product of three trees T1, T2, and T3 defined by three directions of the host hexagonal grid. Namely, to every vertex v of G one can associate an ordered triplet (v1, v2, v3) with vi being a vertex of Ti (i = 1, 2, 3), such that the graph-theoretic distance between two vertices u, v of G equals the sum of respective tree-distances between ui and vi. This labeling of the vertices of G can be obtained in O(n) time. As an application of this result we present an optimal O(n) time algorithm for computing the diameter of the graph G of a benzenoid system with n vertices.

Bridged Graphs Are Cop-Win Graphs

A graph isbridgedif it contains no isometric cycles of length greater than three. Anstee and Farber established that bridged graphs are cop-win graphs. According to Nowakowski and Winkler and Quilliot, a graph is a cop-win graph if and only if its vertices admit a linear orderingv1, v2, ?, vnsuch that every vertexvi,i>1, is dominated by some neighbourvj,j