Tiziana Calamoneri

The L(h, k)-Labelling Problem: A Survey and Annotated Bibliography

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.

The L(h, k)-Labelling Problem

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 a minimum span. The L(h, k)-labelling problem has intensively been 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 previously published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.

L (h,1)-labeling subclasses of planar graphs

L(h, 1)-labeling, h = 0, 1, 2, is a class of coloring problems arising from frequency assignment in radio networks, in which adjacent nodes must receive colors that are at least h apart while nodes connected by a two long path must receive different colors. This problem is NP-complete even when limited to planar graphs. Here, we focus on L(h, 1)-labeling restricted to regular tilings of the plane and to outerplanar graphs. We give a unique parametric algorithm labeling each regular tiling of the plane. For these networks, a channel can be assigned to any node in constant time, provided that relative positions of the node in the network is locally known. Regarding outerplanar graphs with maximum degree Δ, we improve the best known upper bounds from Δ + 9, Δ + 5 and Δ + 3 to Δ + 3, Δ + 1 and Δ colors for the values of h equal to 2, 1 and 0, respectively, for sufficiently large values of Δ. For h = 0, 1 this result proves the polinomiality of the problem for outerplanar graphs. Finally, we study the special case Δ = 3, achieving surprising results.

Push & Pull: autonomous deployment of mobile sensors for a complete coverage

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.

Snap and Spread: A Self-deployment Algorithm for Mobile Sensor Networks

The use of mobile sensors is motivated by the necessity to monitor critical areas where sensor deployment cannot be performed manually. In these working scenarios, sensors must adapt their initial position to reach a final deployment which meets some given performance objectives such as coverage extension and uniformity, total moving distance, number of message exchanges and convergence rate. We propose an original algorithm for autonomous deployment of mobile sensors called Snap & Spread . Decisions regarding the behavior of each sensor are based on locally available information and do not require any prior knowledge of the operating conditions nor any manual tuning of key parameters. We conduct extensive simulations to evaluate the performance of our algorithm. This experimental study shows that, unlike previous solutions, our algorithm reaches a final stable deployment, uniformly covering even irregular target areas. Simulations also give insights on the choice of some algorithm variants that may be used under some different operative settings.

Labeling trees with a condition at distance two

An L(h,k)-labeling of a graph G is an integer labeling of vertices of G, such that adjacent vertices have labels which differ by at least h, and vertices at distance two have labels which differ by at least k. The span of an L(h,k)-labeling is the difference between the largest and the smallest label. We investigate L(h,k)-labelings of trees of maximum degree @D, seeking those with small span. Given @D, h and k, span @l is optimal for the class of trees of maximum degree @D, if @l is the smallest integer such that every tree of maximum degree @D has an L(h,k)-labeling with span at most @l. For all parameters @D,h,k, such that h

3D straight-line grid drawing of 4-colorable graphs

Abstract In this paper we contribute to the understanding of the geometric properties of 3D drawings. Namely, we show how to make a 3D straight-line grid drawing of 4-colorable graphs in O(n2) volume. Moreover, we prove that each bipartite graph needs at least Ω(n 3 2 ) volume.

On relaxing the constraints in pairwise compatibility graphs

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.

All Graphs with at Most Seven Vertices are Pairwise Compatibility Graphs

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Autonomous Deployment of Self-Organizing Mobile Sensors for a Complete Coverage

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