Josep M. Fabrega

Maximally connected digraphs

This paper introduces a new parameter I = I(G) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let k, λ, δ, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that λ = δ if D ⩽ 2I and κ k = δ if D ⩽ 2I - 1. Analogous results involving upper bounds for k and λ are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-λ and super-k are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized.

On the extraconnectivity of graphs

Abstract Given a simple connected graph G , let κ ( n ) [ λ ( n )] be the minimum cardinality of a set of vertices [edges], if any, whose deletion disconnects G and every remaining component has more than n vertices. For instance, the usual connectivity and the superconnectivity of G correspond to κ (0) and κ (1), respectively. This paper gives sufficient conditions, relating the diameter of G with its girth, to assure optimum values of these conditional connectivities.

Next generation sliceable bandwidth variable transponders

This article reports the work on next generation transponders for optical networks carried out within the last few years. A general architecture supporting super-channels (i.e., optical connections composed of several adjacent subcarriers) and sliceability (i.e., subcarriers grouped in a number of independent super-channels with different destinations) is presented. Several transponder implementations supporting different transmission techniques are considered, highlighting advantages, economics, and complexity. Discussions include electronics, optical components, integration, and programmability. Application use cases are reported.

Extraconnectivity of graphs with large girth

Following Harary, the conditional connectivity (edge-connectivity) of a graph with respect to a given graph-theoretic property is the minimum cardinality of a set of vertices (edges), if any, whose deletion disconnects the graph and every remaining component has such a property. We study the case in which all these components are different from a tree whose order is not greater than n. For instance, the recently studied superconnectivity of a maximally connected graph corresponds to this conditional connectivity for n = 1. For other values of n, some sufficient conditions for a graph to have the maximum possible conditional connectivity are given.

Next generation elastic optical networks: The vision of the European research project IDEALIST

In this work we detail the strategies adopted in the European research project IDEALIST to overcome the predicted data plane capacity crunch in optical networks. In order for core and metropolitan telecommunication systems to be able to catch up with Internet traffic, which keeps growing exponentially, we exploit the elastic optical networks paradigm for its astounding characteristics: flexible bandwidth allocation and reach tailoring through adaptive line rate, modulation formats, and spectral efficiency. We emphasize the novelties stemming from the flex-grid concept and report on the corresponding proposed target network scenarios. Fundamental building blocks, like the bandwidth-variable transponder and complementary node architectures ushering those systems, are detailed focusing on physical layer, monitoring aspects, and node architecture design.

Design and Experimental Validation of a GMPLS/PCE Control Plane for Elastic CO-OFDM Optical Networks

ITU-T Recommendation G.694.1 defines normative DWDM frequency grids, each being a reference set of values that correspond to allowed nominal central frequencies, obtained using a fixed channel spacing (e.g., 12.5 GHz, 25 GHz, 50 GHz or 100 GHz). This rigid, grid-based approach does not seem well adapted for data rates beyond 100 Gb/s, is particularly inefficient when a whole wavelength is assigned to a lower rate optical path, and is not flexible enough for multi-rate systems . Consequently, the next generation of optical networks will require a flexible, highly efficient and adaptive management of the optical spectrum, along with advanced optical modulation schemes that efficiently use allocated spectrum slots, and recent progress on optical network technology justifies research on both new optical transmission systems as well as the applicability of control and management frameworks to such networks. We design and deploy a GMPLS control plane for flexible optical networks with coherent optical orthogonal frequency-division multiplexing (O-OFDM) transmission; we detail its functional architecture, which combines a centralized entity that performs path routing and modulation assignment, with a distributed spectrum allocation. The centralized entity (i.e., a path computation element) uses pre-configured static path characterizations, based on exhaustive OFDM transmission simulations, when performing dynamic path computation in line with GMPLS constrained shortest path mechanisms. The distributed spectrum allocation assigns frequency ranges (slots) to connection requests, by using dynamic signaling procedures and applying slot assignment policies. We summarize the control plane protocol extensions involved in the main functional aspects: routing and topology dissemination, path computation, signaling and resource reservation. We experimentally validate and evaluate the integrated centralized PCE and GMPLS control plane in a control plane testbed, obtaining key performance indicators such as path setup latency and blocking probability for different frequency slot assignment policies.

GMPLS/PCE control of flexi-grid DWDM optical networks using CO-OFDM transmission [Invited]

Current optical transport networks use optical channel carriers (wavelengths) that are defined and constrained by a fixed ITU-T dense wavelength division multiplexing (DWDM) grid. Such a grid is not adapted to high data rates (beyond 100 Gb/s) and is inefficient when a wavelength is assigned to a low-rate optical signal. Consequently, the ITU-T is updating the set of DWDM reference frequencies with the inclusion of a smaller channel spacing (e.g., 6.25 GHz) while allowing the allocation of frequency slots, that is, variable-sized frequency ranges composed of a number of slices. In this paper, we propose the design, implementation, and experimental validation of a Generalized Multi-Protocol Label Switching/path computation element (GMPLS/PCE) control plane for such flexible optical networks, using optical orthogonal frequency division multiplexing transmission technology, given its unique flexibility, bit-rate/bandwidth scalability, and subwavelength granularity. The control plane uses a distance-adaptive and PCE-based routing and modulation assignment, combined with distributed frequency slot (spectrum) selection. A comparative analysis of path computation algorithms is carried out, highlighting the benefits of extending the path computation function with the knowledge of the status of the slices and the spectral efficiency of the modulation formats.

On the order and size of s -geodetic digraphs with given connectivity

Abstract A digraph G = (V, E) with diameter D is said to be s-geodetic, for 1 ⩽ s ⩽ D, if between any pair of (not necessarily different) vertices x, y ϵ V there is at most one x → y path of length ⩽ s. Thus, any loopless digraph is at least 1-geodetic. A similar definition applies for a graph G, but in this case the concept is closely related to its girth g, for then G is s-geodetic with s = ⌊(g − 1)/2⌋. The case s = D corresponds to the so-called (strongly) geodetic (di)graphs. Some recent results have shown that if the order n of a (di)graph is big enough, then its vertex connectivity attains its maximum value. In other words, the (di)graph is maximally connected. Moreover, a similar result involving the size m (number of edges) and edge-connectivity applies. In this work we mainly show that the same conclusions can be reached if the order or size of a s-geodetic (di)graph is small enough. As a corollary, we find some Chartrand-type conditions to assure maximum connectivities. For example, when s ⩾ 2, a s-geodetic digraph is maximally connected if δ ⩾ ⌈ 5 n/2−1 ⌉ . Under similar hypotheses it is also shown that stronger measures of connectivity, such as the so-called super-connectivity, attain also their maximum possible values.

Bipartite graphs and digraphs with maximum connectivity

Abstract Recently, some sufficient conditions for a digraph to have maximum connectivity or high superconnectivity have been given in terms of a new parameter which can be thought of as a generalization of the girth of a graph. In this paper similar results are derived for bipartite digraphs and graphs showing that, in this case, all the known conditions can be improved. As a corollary, it is shown that any bipartite graph of girth g and diameter D ⩽ g − 2 (respectively D ⩽ g − 1) has maximum vertex-connectivity (respectively maximum edge-connectivity). This implies a result of Plesnik and Znam stating that any bipartite graph with diameter three is maximally edge-connected.

Extraconnectivity of graphs with large minimum degree and girth

Abstract The extraconnectivity κ ( n ) of a simple connected graph G is a kind of conditional connectivity which is the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has more than n vertices. The usual connectivity and superconnectivity of G correspond to κ (0) and κ (1), respectively. This paper gives sufficient conditions, relating the diameter D , the girth g , and the minimum degree δ of a graph, to assure maximum extraconnectivity. For instance, if D ⩽ g - n + 2( δ - 3), for n ⩾ 2 δ + 4 and g ⩾ n + 5, then the value of κ ( n ) is ( n - 1) δ - 2 n , which is optimal. The corresponding edge version of this result, to assure maximum edge-extraconnectivity λ ( n ), is also discussed.