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 $\delta$-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 $\delta$, i.e. each path is contained in the union of the $\delta$-neighborhoods of two others. A graph $G$ is called $\delta$-slim, if all geodesic triangles in $G$ are $\delta$-slim. The smallest value $\delta$ for which $G$ is $\delta$-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 $\Delta(G)$ of a layering partition of $G$, (2) graphs with tree-length $\lambda$, (3) graphs with tree-breadth $\rho$, (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

Relaxed spanners for directed disk graphs

Let $(V,\delta)$ be a finite metric space, where $V$ is a set of $n$ points and $\delta$ is a distance function defined for these points. Assume that $(V,\delta)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In \cite{PeRo08} we presented an algorithm for constructing a (1+\eps)-spanner of size O(n/\eps^d \log M), where $M$ is the maximal radius $r(p)$. The current paper presents two results. The first shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of $M$. The second result shows that by slightly relaxing the requirements and allowing a small perturbation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph I(V,E,r_{1+\eps}), where r_{1+\eps}(p) = (1+\eps)\cdot r(p) for every $p\in V$, then it is possible to get a (1+\eps)-spanner of size O(n/\eps^d) for $I(V,E,r)$. Our algorithm is simple and can be implemented efficiently

Computing a Minimum-Dilation Spanning Tree is NP-hard

In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices differs from their Euclidean distance. We show that given a set S of n points with integer coordinates in the plane and a rational dilation delta > 1, it is NP-hard to determine whether a spanning tree of S with dilation at most delta exists

New Results on Edge-coloring and Total-coloring of Split Graphs

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a special spanning tree in which the distance between any two adjacent vertices is at most $t$. Given a graph $G$, determining the smallest $t$ for which $G$ is $t$-admissible, i.e. the stretch index of $G$ denoted by $\sigma(G)$, is the goal of the $t$-admissibility problem. Split graphs are $3$-admissible and can be partitioned into three subclasses: split graphs with $\sigma=1, 2$ or $3$. In this work we consider such a partition while dealing with the problem of coloring a split graph. Vizing proved that any graph can have its edges colored with $\Delta$ or $\Delta+1$ colors, and thus can be classified as Class 1 or Class 2, respectively. When both, edges and vertices, are simultaneously colored, i.e., a total coloring of $G$, it is conjectured that any graph can be total colored with $\Delta+1$ or $\Delta+2$ colors, and thus can be classified as Type 1 or Type 2. These both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with $\sigma=2$. For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.Comment: 20 pages, 5 figure

Comment résumer le plan

International audienceCet article concerne les graphes de recouvrement d'un ensemble fini de points du plan Euclidien. Un graphe de recouvrement $H$ est de facteur d'étirement $t$ pour un ensemble de points $S$ si, entre deux points quelconques de $S$, le coût d'un plus court chemin dans $H$ est au plus $t$ fois leur distance Euclidenne. Les graphes de recouvrement d'étirement $t$ (ci-après nommés \emph{$t$-spanneurs}) sont à la base de nombreux algorithmes de routage et de navigation dans le plan. Le graphe (ou triangulation) de Delaunay, le graphe de Gabriel, le graphe de Yao ou le Theta-graphe sont des exemples bien connus de $t$-spanneurs. L'étirement $t$ et le degré maximum des spanneurs sont des paramètres important à minimiser pour l'optimisation des ressources. En même temps le caractère planaire des constructions se révèle essentiel dans les algorithmes de navigation. Nous présentons une série de résultats dans ce domaine, en particulier: \begin{itemize} \item Nous montrons que le graphe $\Theta_6$ (le Theta-graphe où $k=6$ cônes d'angle $\Theta_k = 2\pi/k$ par sommet sont utilisées) est l'union de deux spanneurs planaires d'étirement deux. En particulier, nous établissons que l'étirement maximum du graphe $\Theta_6$ est deux, ce qui est optimal. Des bornes supérieures sur l'étirement du graphe $\Theta_k$ n'étaient connues que lorsque $k > 6$. Pour $k=7$, la meilleure borne connue est d'environ $7.56$ et pour $k=6$ il était ouvert de savoir si le graphe était un $t$-spanneur pour une valeur constante de $t$. \item Nous montrons que le graphe $\Theta_6$ contient comme sous-graphe couvrant un $3$-spanneur planaire de degré maximum au plus~$9$. \item Finalement, en utilisant une variante du résultat précédant, nous montrons que le plan Euclidien possède un $6$-spanneur planaire de degré maximum au plus~$6$. \end{itemize} La dernière construction, non décrite ici par manque de place, améliore une longue série de résultats sur le problème largement ouvert de déterminer la plus petite valeur $\delta$ telle que tout ensemble du plan possède un spanneur planaire d'étirement constant et de degré maximum $\delta$. Le meilleur résultat en date montrait que $3 \le \delta\le 14$

Quantum speedup for graph sparsification, cut approximation, and Laplacian solving

Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, “spectral sparsification” reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with n nodes and m edges, outputs a classical description of an ϵ -spectral sparsifier in sublinear time O˜(mn−−−√/ϵ) . This contrasts with the optimal classical complexity O˜(m) . We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for k -wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut

Geometric Dilation and Halving Distance

Let us consider the network of streets of a city represented by a geometric graph G in the plane. The vertices of G represent the crossroads and the edges represent the streets. The latter do not have to be straight line segments, they may be curved. If one wants to drive from a place p to some other place q, normally the length of the shortest path along streets, d_G(p,q), is bigger than the airline distance (Euclidean distance) |pq|. The (relative) DETOUR is defined as delta_G(p,q) := d_G(p,q)/|pq|. The supremum of all these ratios is called the GEOMETRIC DILATION of G. It measures the quality of the network. A small dilation value guarantees that there is no bigger detour between any two points. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the DILATION of S and denote it by delta(S). The main results of this thesis are - a general upper bound to the dilation of any finite point set S, delta(S) - a lower bound for a specific set P, delta(P)>(1+10^(-11))pi/2, which approximately equals 1.571 In order to achieve these results, we first consider closed curves. Their dilation depends on the HALVING PAIRS, pairs of points which divide the closed curve in two parts of equal length. In particular the distance between the two points is essential, the HALVING DISTANCE. A transformation technique based on halving pairs, the HALVING PAIR TRANSFORMATION, and the curve formed by the midpoints of the halving pairs, the MIDPOINT CURVE, help us to derive lower bounds to dilation. For constructing graphs of small dilation, we use ZINDLER CURVES. These are closed curves of constant halving distance. To give a structured overview, the mathematical apparatus for deriving the main results of this thesis includes - upper bound: * the construction of certain Zindler curves to generate a periodic graph of small dilation * an embedding argument based on a number theoretical result by Dirichlet - lower bound: * the formulation and analysis of the halving pair transformation * a stability result for the dilation of closed curves based on this transformation and the midpoint curve * the application of a disk-packing result In addition, this thesis contains - a detailed analysis of the dilation of closed curves - a collection of inequalities, which relate halving distance to other important quantities from convex geometry, and their proofs; including four new inequalities - the rediscovery of Zindler curves and a compact presentation of their properties - a proof of the applied disk packing result.Geometrische Dilation und Halbierungsabstand Man kann das von den Straßen einer Stadt gebildete Netzwerk durch einen geometrischen Graphen in der Ebene darstellen. Die Knoten dieses Graphen repräsentieren die Kreuzungen und die Kanten sind die Straßen. Letztere müssen nicht geradlinig sein, sondern können beliebig gekrümmt sein. Wenn man nun von einem Ort p zu einem anderen Ort q fahren möchte, dann ist normalerweise die Länge des kürzesten Pfades über Straßen, d_G(p,q), länger als der Luftlinienabstand (euklidischer Abstand) |pq|. Der (relative) UMWEG (DETOUR) ist definiert als delta_G(p,q) := d_G(p,q)/|pq|. Das Supremum all dieser Brüche wird GEOMETRISCHE DILATION (GEOMETRIC DILATION) von G genannt. Es ist ein Maß für die Qualität des Straßennetzes. Ein kleiner Dilationswert garantiert, dass es keinen größeren Umweg zwischen beliebigen zwei Punkten gibt. Für eine gegebene endliche Punktmenge S würden wir nun gerne bestimmen, was der kleinste Dilationswert ist, den wir mit einem Graphen erreichen können, der die gegebenen Punkte auf seinen Kanten enthält. Dieses Infimum nennen wir die DILATION von S und schreiben kurz delta(S). Die Haupt-Ergebnisse dieser Arbeit sind - eine allgemeine obere Schranke für die Dilation jeder beliebigen endlichen Punktmenge S: delta(S) - eine untere Schranke für eine bestimmte Menge P: delta(P)>(1+10^(-11))pi/2, was ungefähr der Zahl 1.571 entspricht Um diese Ergebnisse zu erreichen, betrachten wir zunächst geschlossene Kurven. Ihre Dilation hängt von sogenannten HALBIERUNGSPAAREN (HALVING PAIRS) ab. Das sind Punktpaare, die die geschlossene Kurve in zwei Teile gleicher Länge teilen. Besonders der Abstand der beiden Punkte ist von Bedeutung, der HALBIERUNGSABSTAND (HALVING DISTANCE). Eine auf den Halbierungspaaren aufbauende Transformation, die HALBIERUNGSPAARTRANSFORMATION (HALVING PAIR TRANSFORMATION), und die von den Mittelpunkten der Halbierungspaare gebildete Kurve, die MITTELPUNKTKURVE (MIDPOINT CURVE), helfen uns untere Dilationsschranken herzuleiten. Zur Konstruktion von Graphen mit kleiner Dilation benutzen wir ZINDLERKURVEN (ZINDLER CURVES). Dies sind geschlossene Kurven mit konstantem Halbierungspaarabstand. Die mathematischen Hilfsmittel, mit deren Hilfe wir schließlich die Hauptresultate beweisen, sind unter anderem - obere Schranke: * die Konstruktion von bestimmten Zindlerkurven, mit denen periodische Graphen kleiner Dilation gebildet werden können * ein Einbettungsargument, das einen zahlentheoretischen Satz von Dirichlet benutzt - untere Schranke: * die Definition und Analyse der Halbierungspaartransformation * ein Stabilitätsresultat für die Dilation geschlossener Kurven, das auf dieser Transformation und der Mittelpunktkurve basiert * die Anwendung eines Kreispackungssatzes Zusätzlich enthält diese Dissertation - eine detaillierte Analyse der Dilation geschlossener Kurven - eine Sammlung von Ungleichungen, die den Halbierungsabstand zu anderen wichtigen Größen der Konvexgeometrie in Beziehung setzen, und ihre Beweise; inklusive vier neuer Ungleichungen - die Wiederentdeckung von Zindlerkurven und eine kompakte Darstellung ihrer Eigenschaften - einen Beweis des angewendeten Kreispackungssatzes

Rapid onset of molecular friction in liquids bridging between the atomistic and hydrodynamic pictures

Friction in liquids arises from conservative forces between molecules and atoms. Although the hydrodynamics at the nanoscale is subject of intense research and despite the enormous interest in the non-Markovian dynamics of single molecules and solutes, the onset of friction from the atomistic scale so far could not be demonstrated. Here, we fill this gap based on frequency-resolved friction data from high-precision simulations of three prototypical liquids, including water. Combining with theory, we show that friction in liquids emerges abruptly at a characteristic frequency, beyond which viscous liquids appear as non-dissipative, elastic solids. Concomitantly, the molecules experience Brownian forces that display persistent correlations. A critical test of the generalised Stokes–Einstein relation, mapping the friction of single molecules to the visco-elastic response of the macroscopic sample, disproves the relation for Newtonian fluids, but substantiates it exemplarily for water and a moderately supercooled liquid. The employed approach is suitable to yield insights into vitrification mechanisms and the intriguing mechanical properties of soft materials