Given any fixed nonnegative integer values h and k, the L(h, k)labelling problem consists in an assignment of nonnegative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with minimum span. The L(h, k)-labelling problem has been intensively studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previous by published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.

Given any fixed non-negative integer values h and k, the L(h, k)-labelling problem consists in an assignment of non-negative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with minimum span. The L(h, k)-labelling problem has been intensively studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previous by published literature, looking at the problem with a graph algorithmic approach.

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In this paper, we address the problem of deploying heterogeneous mobile sensors over a target area. Traditional approaches to mobile sensor deployment are specifically designed for homogeneous networks. Nevertheless, network and device homogeneity is an unrealistic assumption in most practical scenarios, and previous approaches fail when adopted in heterogeneous operative settings. For this reason, we introduce VorLag, a generalization of the Voronoi-based approach which exploits the Laguerre geometry. We theoretically prove the appropriateness of our proposal to the management of heterogeneous networks. In addition, we demonstrate that VorLag can be extended to deal with dynamically generated events or uneven energy depletion due to communications. Finally, by means of simulations, we show that VorLag provides a very stable sensor behavior, with fast and guaranteed termination and moderate energy consumption. We also show that VorLag performs better than its traditional counterpart and other methods based on virtual forces

Mobile sensor networks are important for several strategic applications devoted to monitoring critical areas. In such hostile scenarios, sensors cannot be deployed manually and are either sent from a safe location or dropped from an aircraft. Mobile devices permit a dynamic deployment reconfiguration that improves the coverage in terms of completeness and uniformity. In this paper we propose a distributed algorithm for the autonomous deployment of mobile sensors called Push & Pull. According to our proposal, movement decisions are made by each sensor on the basis of locally available information and do not require any prior knowledge of the operating conditions or any manual tuning of key parameters. We formally prove that, when a sufficient number of sensors are available, our approach guarantees a complete and uniform coverage. Furthermore, we demonstrate that the algorithm execution always terminates preventing movement oscillations. Numerous simulations show that our algorithm reaches a complete coverage within reasonable time with moderate energy consumption, even when the target area has irregular shapes. Performance comparisons between Push & Pull and one of the most acknowledged algorithms show how the former one can efficiently reach a more uniform and complete coverage under a wide range of working scenarios

A graph G = (V, E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers d min and d max such that each leaf u of T is a node of V and there is an edge (u, v) ∈ E if and only if d min ≤ d T (u, v) ≤ d max where d T (u, v) is the sum of weights of the edges on the unique path from u to v in T. In this paper, we survey the state of the art concerning this class of graphs and some of its subclasses.

A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers dmin and dmax such that each leaf lu of T corresponds to a vertex u ∈ V and there is an edge (u, v) ∈ E if and only if dmin ≤ dT (lu, lv) ≤ dmax where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T . In this paper we analyze the class of PCG in relation with two particular subclasses resulting from the the cases where dmin = 0 (LPG) and dmax = +∞ (mLPG). In particular, we show that the union of LPG and mLPG does not coincide with the whole class PCG, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, as the graphs we deal with belong to the more general class of split matrogenic graphs, we focus on this class of graphs for which we try to establish the membership to the PCG class.

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