(Nearly-)tight bounds on the contiguity and linearity of cographs

International audienceIn this paper we show that the contiguity and linearity of cographs on n vertices are both O(n). Moreover, we show that this bound is tight for contiguity as there exists a family of cographs on n vertices whose contiguity is Ω(log n). We also provide an Ω(log n / log log n) lower bound on the maximum linearity of cographs on n vertices. As a by-product of our proofs, we obtain a min-max theorem, which is worth of interest in itself, stating equality between the rank of a tree and the minimum height of one of its path partitions

Linear-time Constant-ratio Approximation Algorithm and Tight Bounds for the Contiguity of Cographs

International audienceIn this paper we consider a graph parameter called contiguity which aims at encoding a graph by a linear ordering of its vertices. We prove that the contiguity of cographs is unbounded but is always dominated by O(log n), where n is the number of vertices of the graph. And we prove that this bound is tight in the sense that there exists a family of cographs on n vertices whose contiguity is Omega(log n). In addition to these results on the worst-case contiguity of cographs, we design a linear-time constant-ratio approximation algorithm for computing the contiguity of an arbitrary cograph, which constitutes our main result. As a by-product of our proofs, we obtain a min-max theorem, which is worth of interest in itself, stating equality between the rank of a tree and the minimum height of its path partitions

On the effectiveness of the incremental approach to minimal chordal edge modification

Because edge modification problems are computationally difficult for most target graph classes, considerable attention has been devoted to inclusion-minimal edge modifications, which are usually polynomial-time computable and which can serve as an approximation of minimum cardinality edge modifications, albeit with no guarantee on the cardinality of the resulting modification set. Over the past fifteen years, the primary design approach used for inclusion-minimal edge modification algorithms is based on a specific incremental scheme. Unfortunately, nothing guarantees that the set E of edge modifications of a graph G that can be obtained in this specific way spans all the inclusion-minimal edge modifications of G. Here, we focus on edge modification problems into the class of chordal graphs and we show that for this the set E may not even contain any solution of minimum size and may not even contain a solution close to the minimum; in fact, we show that it may not contain a solution better than within an Ω(n) factor of the minimum. These results show strong limitations on the use of the current favored algorithmic approach to inclusion-minimal edge modification in heuristics for computing a minimum cardinality edge modification. They suggest that further developments might be better using other approaches.publishedVersio

Rigorous Measurement of IP-Level Neighborhood of Internet Core Routers

International audienceMany contributions use the degree distribution of IP-level internet topology. However, current knowledge of this property relies on biased and erroneous measurements, and so it is subject to much debate. We introduce here a new approach, dedicated to the core of the internet, which avoids the issues raised by classical measurements. It is based on the measurement of IP-level neighborhood of internet core routers, for which we design and implement a rigorous method. It consists in sending traceroute probes from many monitors distributed in the internet towards a given target router and carefully selecting the relevant information in collected data. Using simulations, we provide strong evidence of the accuracy of our approach. We then conduct real-world measurements illustrating the practical effectiveness of our method. This constitutes a significant step towards reliable knowledge of the IP-level degree distribution of the core of the internet

Measuring routing tables in the internet

International audienceThe most basic function of an Internet router is to decide, for a given packet, which of its interfaces it will use to forward it to its next hop. To do so, routers maintain a routing table, in which they look up for a prefix of the destination address. The routing table associates an interface of the router to this prefix, and this interface is used to forward the packet. We explore here a new measurement method based upon distributed UDP probing to estimate this routing table for Internet routers

Cyclability in Graph Classes

A subset T subseteq V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every subset of vertices of G of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is, a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes

On the termination of some biclique operators on multipartite graphs

International audienceWe define a new graph operator, called the weak-factor graph, which comes from the context of complex network modelling. The weak-factor operator is close to the well-known clique-graph operator but it rather operates in terms of bicliques in a multipartite graph. We address the problem of the termination of the series of graphs obtained by iteratively applying the weak-factor operator starting from a given input graph. As for the clique-graph operator, it turns out that some graphs give rise to series that do not terminate. Therefore, we design a slight variation of the weak-factor operator, called clean-factor, and prove that its associated series terminates for all input graphs. In addition, we show that the multipartite graph on which the series terminates has a very nice combinatorial structure: we exhibit a bijection between its vertices and the chains of the inclusion order on the intersections of the maximal cliques of the input graph

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio