Consider a team of mobile software agents deployed to capture a (possibly hostile) intruder in a network. All agents, including the intruder move along the network links; the intruder could be arbitrarily fast, and aware of the positions of all the agents. The problem is to design the agents' strategy for capturing the intruder. The main efficiency parameter is the size of the team. This is an instance of the well known graph-searching problem whose many variants have been extensively studied in the literature. In all existing solutions, and in all the variants of the problem, it is assumed that agents can be removed from their current location and placed in another network site arbitrarily and at any time. As a consequence, the existing optimal strategies cannot be employed in situations for which agents cannot access the network at any point, or cannot "jump" across the network, or cannot reach an arbitrary point of the network via an internal travel through insecure zones. This motivates the contiguous search problem in which agents cannot be removed from the network, and clear links must form a connected sub-network at any time, providing safety of movements. This new problem is NP-complete in general. We study it for tree networks, and we consider its more general version, the weighted case, which arises naturally when considering networks whose nodes and links are of different nature and thus require a different number of agents to be explored. We give a linear-time algorithm that computes, for any tree T , the minimum number of agents to capture the intruder, and the corresponding search strategy. Beside its optimality in time, our algorithm is naturally distributed:

Several papers showed how to perform routing in ad hoc wireless networks based on the positions of the mobile hosts. However, all these protocols are likely to fall if the transmission ranges of the mobile hosts vary due to natural or man-made obstacles or weather conditions. These protocols may fall because in routing either some connections are not considered which effectively results in disconnecting the network, or the use of some connections causes livelocks. In this paper, we describe a robust routing protocol that tolerates up to roughly 40% of variation in the transmission ranges of the mobile hosts. More precisely, our protocol guarantees message delivery in a connected adhoc network whenever the ratio of the maximum transmission range to the minimum transmission range is at most V~. General Terms Mobile computing and Communications

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A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some well-known properties of the cartesian product, such as a reduced mean distance and diameter, simple routing algorithms and some optimal communication protocols are inherited by the hierarchical product. We also address the study of some algebraic properties of the hierarchical product of two or more graphs. In particular, the spectrum of the binary hypertree T m (which is the hierarchical product of several copies of the complete graph on two vertices) is fully characterized; turning out to be an interesting example of graph with all its eigenvalues distinct. Finally, some natural generalizations of the hierarchic product are proposed.

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We consider the uniform scattering problem for a set of autonomous mobile robots deployed in a grid network: starting from an arbitrary placement in the grid, using purely localized computations, the robots must move so to reach in finite time a state of static equilibrium in which they cover uniformly the grid. The theoretical quest is on determining the minimal capabilities needed by the robots to solve the problem. We prove that uniform scattering is indeed possible even for very weak robots. The proof is constructive. We present a provably correct protocol for uniform self-deployment in a grid. The protocol is fully localized, collision-free, and it makes minimal assumptions; in particular: (1) it does not require any direct or explicit communication between robots; (2) it makes no assumption on robots synchronization or timing, hence the robots can be fully asynchronous in all their actions; (3) it requires only a limited visibility range; (4) it uses at each robot only a constant size memory, hence computationally the robots can be simple Finite-State Machines; (5) it does not need a global localization system but only orientation in the grid (e.g., a compass); (6) it does not require identifiers, hence the robots can be anonymous and totally identical.

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.

Consider a collection of r identical asynchronous mobile agents dispersed on an arbitrary anonymous network of size n. The agents all execute the same protocol and move from node to neighbouring node. At each node there is a whiteboard where the agents can write and read from. The topology of the network is unknown to the agents. We examine the problems of rendezvous (i.e., having the agents gather in the same node) and election (i.e., selecting a leader among those agents). These two problems are computationally equivalent in the context examined here. We study conditions for the existence of deterministic generic solutions, i.e., algorithms that solve the two problems regardless of the network topology and the initial placement of the agents. In particular, we study the impact of edge-labeling on the existence of such solutions. Rendezvous and election are unsolvable (i.e., there are no deterministic generic solutions) if gcd(r, n) > 1, regardless of whether or not the edge-labeling has sense of direction. On the other hand, if gcd(r, n) = 1 then the initial placement of the robots in the network creates topological asymmetries that could be exploited to solve the problems. We prove that these asymmetries can be exploited if the edge labeling has sense of direction, but can not if the edge-labeling is arbitrary. The possibility proof is constructive: we present a solution protocol and prove its correctness. The protocol, among other features, uses a dynamic naming mechanism based on sense of direction to overcome the complete anonymity of the system.