Nathann Cohen

On Computing the Gromov Hyperbolicity

The Gromov hyperbolicity is an important parameter for analyzing complex networks which expresses how the metric structure of a network looks like a tree. It is for instance used to provide bounds on the expected stretch of greedy-routing algorithms in Internet-like graphs. However, the best-known theoretical algorithm computing this parameter runs in O(n3.69) time, which is prohibitive for large-scale graphs. In this article, we propose an algorithm for determining the hyperbolicity of graphs with tens of thousands of nodes. Its running time depends on the distribution of distances and on the actual value of the hyperbolicity. Although its worst case runtime is O(n4), it is in practice much faster than previous proposals as observed in our experimentations. Finally, we propose a heuristic algorithm that can be used on graphs with millions of nodes. Our algorithms are all evaluated on benchmark instances.

Coloring hypergraphs induced by dynamic point sets and bottomless rectangles

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k−2. This can be interpreted as coloring point sets in ℝ2 with k colors such that any bottomless rectangle containing at least 3k−2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)>ck, where c>1.67. Hence, for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).

Graphs with maximal irregularity

Albertson (3) has defined the irregularity of a simple undirected graph G = (V; E) as irr(G) =

Acyclic edge-colouring of planar graphs. Extended abstract

Abstract A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. The acyclic chromatic index of a graph G, denoted χ a ′ ( G ) is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and Δ ( G ) is large enough then χ a ′ ( G ) = Δ ( G ) . We settle this conjecture for planar graphs with girth at least 5 and outerplanar graphs. We also show that if G is planar then χ a ′ ( G ) ⩽ Δ ( G ) + 25 .

A note on Zagreb indices inequality for trees and unicyclic graphs

For a simple graph G with n vertices and m edges, the inequality M 1 ( G )/ n ≤ M 2 ( G )/ m , where M 1 ( G ) and M 2 ( G ) are the first and the second Zagreb indices of G , is known as Zagreb indices inequality. Recently Vukicevic and Graovac [VG], and Caporossi, Hansen and Vukcevic [CHV] proved that this inequality holds for trees and unicyclic graphs, respectively. Here, alternative and shorter proofs of these results are presented. [VG] D. Vukicevic and A. Graovac, Comparing Zagreb M 1 and M 2 indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57 (2007), 587-590. [CHV] G. Caporossi, P. Hansen and D. Vukicevic, Comparing Zagreb indices of cyclic graphs, MATCH Commun. Math. Comput. Chem. 63 (2010), 441-451.

Finding good 2-partitions of digraphs II. Enumerable properties

We continue the study, initiated in 3, of the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties and given minimum cardinality. Let E be the following set of properties of digraphs: E = {strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper we determine, for all choices of P 1 , P 2 from E and all pairs of fixed positive integers k 1 , k 2 , the complexity of deciding whether a digraph has a vertex partition into two digraphs D 1 , D 2 such that D i has property P i and | V ( D i ) | ź k i , i = 1 , 2 . We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the analogous problems when P 1 ź H and P 2 ź H ź E , where H = {acyclic, complete, arc-less, oriented (no 2-cycle), semicomplete, symmetric, tournament} were completely characterized in 3.

Tradeoffs in process strategy games with application in the WDM reconfiguration problem

We consider a variant of the graph searching games that models the routing reconfiguration problem in WDM networks. In the digraph processing game, a team of agents aims at processing, or clearing, the vertices of a digraph D. We are interested in two different measures: (1) the total number of agents used, and (2) the total number of vertices occupied by an agent during the processing of D. These measures, respectively, correspond to the maximum number of simultaneous connections interrupted and to the total number of interruptions during a routing reconfiguration in a WDM network. Previous works have studied the problem of independently minimizing each of these parameters. In particular, the corresponding minimization problems are APX-hard, and the first one is known not to be in APX. In this paper, we give several complexity results and study tradeoffs between these conflicting objectives. In particular, we show that minimizing one of these parameters while the other is constrained is NP-complete. Then, we prove that there exist some digraphs for which minimizing one of these objectives arbitrarily impairs the quality of the solution for the other one. We show that such bad tradeoffs may happen even for a basic class of digraphs. On the other hand, we exhibit classes of graphs for which good tradeoffs can be achieved. We finally detail the relationship between this game and the routing reconfiguration problem. In particular, we prove that any instance of the processing game, i.e. any digraph, corresponds to an instance of the routing reconfiguration problem.

Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable-A short proof

We give a short proof of the following theorem due to Borodin (1990) [2]. Every planar graph G with maximum degree at least 9 is (@D(G)+1)-edge-choosable.

Implementing Brouwer's database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on