A set C of vertices in a graph G = (V, E) is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala’s contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes.

In an optical network, a Scheduled Lightpath Demand (SLD) is a connection demand between two nodes, during a certain time and with a certain wavelength. We consider the following NP-hard Routing and Wavelength Assignment problem dealing with SLDs: given a set of SLDs and a number W of wavelengths, maximize the number of SLDs to which we can assign a lightpath (i.e. a routing path and a wavelength) without exceeding the number W of available wavelengths. The constraints are: a same wavelength must be assigned all along the routing path of any SLD; at any time, a given wavelength on a given edge of the network cannot be used to satisfy more than one SLD. To solve this problem, we study an approach stating the problem as the successive searches of independent sets in some conflict graphs. Moreover, we improve this approach thanks to a post-optimization method. The experimental results show that this model and the post-optimization method are quite efficient to provide a large number of routed SLDs.

Wireless sensor networks have been widely deployed in the last decades to provide various services, like environmental monitoring or object tracking. Such a network is composed of a set of sensor nodes which are used to sense and transmit collected information to a base station. To achieve this goal, two properties have to be guaranteed: (i) the sensor nodes must be placed such that all the environment of interest is covered, and (ii) every sensor node can transmit its data to the base station (through other sensor nodes). In this paper, we consider the Minimum Connected Coverage (MCC) problem. We propose two mathematical programming formulations for the MCC problem on square grid graphs. We compare them to a recent model proposed by Rebai et al [1]. Our mathematical programming formulations yield a better LP-bound at the root of the branch-and-cut process than the model of Rebai et al. Moreover, the presented formulations outperform the proportion of solved instances in their work as well as the CPU computation time and the number of nodes explored in the tree search.

Let G=(V, A) be a directed, asymmetric graph and C a subset of vertices, and let B/sub r//sup -/(v) denote the set of all vertices x such that there exists a directed path from x to v with at most r arcs. If the sets B/sub r//sup -/(v) /spl cap/ C, v /spl isin/ V (respectively, v /spl isin/ V/spl bsol/C), are all nonempty and different, we call C an r-identifying code (respectively, an r-locating-dominating code) of G. In other words, if C is an r-identifying code, then one can uniquely identify a vertex v /spl isin/ V only by knowing which codewords belong to B/sub r//sup -/(v), and if C is r-locating-dominating, the same is true for the vertices v in V/spl bsol/C. We prove that, given a directed, asymmetric graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r/spl ges/1 and remains so even when restricted to strongly connected, directed, asymmetric, bipartite graphs or to directed, asymmetric, bipartite graphs without directed cycles.